What is the Biblical basis for saying that we won't have a second chance to be saved after we die?

Things become straightforward if we go back to the only biblically wholesome teaching about the afterlife, which is the general resurrection either to life everlasting, for all those whose name was "written in the book of life" or to be thrown in the "lake of fire" for all all those whose name "was not found written in the book of life" (Rev 20:11-14 NET).

When would things be written about us in the "book of life"? Well quite obviously ... during our life.

Not only "purgatory" but also "heaven" and "hell", as some sort of "places" to accomodate for the "immortal soul" after death, are mere priestly inventions, concessions to the superstitions of the populace and distortions of Biblical doctrine with the help of heathen philosophy.

The bottom line?

We won't have a second chance to be saved after we die!

We must put our lives in order while we are ... alive.

For those who believe you may only be saved before your death, and not after, there are two Biblical verses that may apply.

Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. John 3:5

The Catholic Church mentions this in the first footnote "60", but expands the meaning of "born of water and Spirit" to include water, blood, desire, God's way, and God's mercy.

Water baptism, blood baptism, desire baptism, God's way, and God's mercy

VI. THE NECESSITY OF BAPTISM

1257 The Lord himself affirms that Baptism is necessary for salvation.60 [quoted above] He also commands his disciples to proclaim the Gospel to all nations and to baptize them.61 Baptism is necessary for salvation for those to whom the Gospel has been proclaimed and who have had the possibility of asking for this sacrament.62 The Church does not know of any means other than Baptism that assures entry into eternal beatitude; this is why she takes care not to neglect the mission she has received from the Lord to see that all who can be baptized are "reborn of water and the Spirit." God has bound salvation to the sacrament of Baptism, but he himself is not bound by his sacraments.

1258 The Church has always held the firm conviction that those who suffer death for the sake of the faith without having received Baptism are baptized by their death for and with Christ. This Baptism of blood, like the desire for Baptism, brings about the fruits of Baptism without being a sacrament.

1259 For catechumens who die before their Baptism, their explicit desire to receive it, together with repentance for their sins, and charity, assures them the salvation that they were not able to receive through the sacrament.

1260 "Since Christ died for all, and since all men are in fact called to one and the same destiny, which is divine, we must hold that the Holy Spirit offers to all the possibility of being made partakers, in a way known to God, of the Paschal mystery."63 Every man who is ignorant of the Gospel of Christ and of his Church, but seeks the truth and does the will of God in accordance with his understanding of it, can be saved. It may be supposed that such persons would have desired Baptism explicitly if they had known its necessity.

1261 As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus' tenderness toward children which caused him to say: "Let the children come to me, do not hinder them,"64 allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church's call not to prevent little children coming to Christ through the gift of holy Baptism. CCC -source- (emphasis mine)

Another Biblical verse that suggests one must believe when alive, before dying, is this.

And as it is appointed unto men once to die, but after this the judgment: Heb 9:27

Apparently you die, then are judged as to your salvation.

So, to answer the OP, the Biblical basis to only be saved prior to your death is the necessity to be born again of water and Spirit, and the verse about dying and then judgment.

PS. Why can't you be baptized after death?

Now this I say, brethren, that flesh and blood cannot inherit the kingdom of God; neither doth corruption inherit incorruption. 1 Cor 15:50

What exactly would be baptized? You will have some type of body, but is it baptizable? Or has the point been missed, which is to say it is your spirit that is born-again.

Yet, the verse remains that one dies, and one is judged (Heb 9:27).

What is the Biblical basis for saying that we won't have a second chance to be saved after we die?

There is no explicit basis for this in the Bible. On the other hand, there is nothing in the Bible that implies that anyone would be given a second chance.

Everyone has one chance. If they then choose to reject God, to no longer listen to God's holy spirit, that is the unpardonable sin, and they will be condemned to destruction and permanent death.

But, the vast majority of people never had their first chance:

• Jesus is the only name by which salvation is offered, and that wasn't available until 2,000 years ago.
• Until recently, most people in the world lived in Asia, Africa, and the Americas and never had an opportunity to know anything about the God of the Bible.
• Even recently, most people that have heard about Jesus and salvation actually heard about a false Jesus and a false gospel. (E.g. Jehovah's Witnesses, Latter Day Saints, and Roman Catholics each believe that while the other two groups may be sincere in their beliefs, they are sincerely wrong.)
• God wants everyone to be saved.
• Jesus predicted that his Church would be a small flock.

Does it really make sense that God would have created a hundred billion people, never have given them a chance at salvation, and then deliberately subject them to continuous torture and pain for all eternity? That is not the "God of love" described in my Bible; that's a sadistic psychopath.

That small flock, the "elect", will be resurrected at the beginning of the Millennium when Christ returns to Earth, and they will rule with him for a thousand years (the first general resurrection).

But (and here's the part that matches the question's "idea that we can still make a decision for Christ after this life"), at the end of the Millennium there will be a second general resurrection of everyone that never had their chance at salvation.

Mention of this concept typically results in people claiming it is about a second chance, but it isn't. It is their first chance.

For more specifics about these resurrections, see my answer to What is the Order of the Resurrection of the Dead?.

For an example of one (of many) denomination that teaches this doctrine, see: Bible Prophecy and You: The Second Resurrection: Humanity’s Opportunity for Salvation! | United Church of God

Adding to David Stratton's very good answer, consider also that there is no instance recorded in the Scriptures of a person being saved after death. Ananias and Sapphira would be good candidates for this, if it were available, but we read of no such thing.

This leads to one of two conclusions: Either there simply will be no opportunity after death for salvation, or we are to not place our hopes in such.

This makes the hardest sense. Let us imagine for a moment that we were going to receive another chance after we have died. Would anybody repent prior to this time? I doubt even one in a million would; we would indulge our passions to whatever degree pleased us, allowing only the consequences in this life to rein in our impulses, and defer all thought of God, kidding ourselves with the notion that we can fix it all later.

The lure of this teaching is clear. Instead of having to make the choice between God and the world now—when we can see what we have to give up, but can only hope for what we are to gain—it tells us that the choice can be put off until a time when we can see what there is to be gained by choosing God, and what will be lost by choosing Him will already have been lost.

In short, we would live as if there were no God.

After a time we would become so corrupt that we would not want to make the choices required for salvation, and we would be lost, even with the judgment staring us in the face.

The biblical basis for saying that we won't have a second chance to be saved after we die is that the righteously default condition of every person before God is condemnation for having rejected the light that they were given:

For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. For God sent not his Son into the world to condemn the world; but that the world through him might be saved. He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God. And this is the condemnation, that light is come into the world, and men loved darkness rather than light, because their deeds were evil. For every one that doeth evil hateth the light, neither cometh to the light, lest his deeds should be reproved. But he that doeth truth cometh to the light, that his deeds may be made manifest, that they are wrought in God. - John 3:16-21

Jesus Christ, the Son of God, came into a world that is already under the just condemnation of God in order that those who believe might be saved through him. This sacrifice is pure mercy and love which God was under no judicial imperative to provide. It would not have been judicially "wrong" for God to consummate His righteous judgement of the condemnation of all rather than institute a means of salvation because all have actively sinned.

As it is written, There is none righteous, no, not one: There is none that understandeth, there is none that seeketh after God. They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one.  Their throat is an open sepulchre; with their tongues they have used deceit; the poison of asps is under their lips: Whose mouth is full of cursing and bitterness: Their feet are swift to shed blood: Destruction and misery are in their ways: And the way of peace have they not known: There is no fear of God before their eyes. Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. - Romans 3:10-19

• Note: The condition of those who die unborn and those before an "age of accountability" are a different matter as these are usually described as "covered" rather than "in need of covering".

In order to remain Just and also justify the ungodly God sent His Son to die for them. This does not indicate Injustice prior to the advent of Christ, prior to the coming of the promised Messiah. It is not unjust for God to condemn but, apart from Christ, it would be unjust for God to forgive:

For all have sinned, and come short of the glory of God; Being justified freely by his grace through the redemption that is in Christ Jesus: Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God; To declare, I say, at this time his righteousness: that he might be just, and the justifier of him which believeth in Jesus. Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith. Therefore we conclude that a man is justified by faith without the deeds of the law. - Romans 3:23-28

There is no sense in which some vast multitude of long ago had no light of truth to guide them unto salvation by grace through faith and will receive a first chance to believe after death. The Bible is clear that creation itself testifies of God's glory and that God's moral law is written in the hearts of all, even those who do not have the formal law:

For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law; (For not the hearers of the law are just before God, but the doers of the law shall be justified. For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves: Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another;) In the day when God shall judge the secrets of men by Jesus Christ according to my gospel. - Romans 2:12-16

When we respond in faith to what light (of truth) we have been given we are given more light. It is not for us to judge that the lost pygmy tribe in the jungles of the Amazon have never received any light when Scripture flat out says that they have. Those who have been inwardly responding to the light they have will respond to the greater light of Christ also:

Thy mercy, O LORD, is in the heavens; and thy faithfulness reacheth unto the clouds. Thy righteousness is like the great mountains; thy judgments are a great deep: O LORD, thou preservest man and beast. How excellent is thy lovingkindness, O God! therefore the children of men put their trust under the shadow of thy wings. They shall be abundantly satisfied with the fatness of thy house; and thou shalt make them drink of the river of thy pleasures. For with thee is the fountain of life: in thy light shall we see light. - Psalm 36:5-9

Old testament saints, prior to the advent of Christ, lived a life of faith looking forward to the fulfillment of God's promise of a savior, first given to Adam and Eve in the garden. Abraham saw Jesus' day and was glad. They had light enough to believe and to be justified by faith in the Savior who was to come. Abraham believed and that belief was credited to his account before God as righteousness:

After these things the word of the LORD came unto Abram in a vision, saying, Fear not, Abram: I am thy shield, and thy exceeding great reward. And Abram said, Lord GOD, what wilt thou give me, seeing I go childless, and the steward of my house is this Eliezer of Damascus? And Abram said, Behold, to me thou hast given no seed: and, lo, one born in my house is mine heir. And, behold, the word of the LORD came unto him, saying, This shall not be thine heir; but he that shall come forth out of thine own bowels shall be thine heir. And he brought him forth abroad, and said, Look now toward heaven, and tell the stars, if thou be able to number them: and he said unto him, So shall thy seed be. And he believed in the LORD; and he counted it to him for righteousness. - Genesis 15:1-6

It can easily be deduced from Scripture that each moment of life, during which creation continually declares the Glory of God, constitutes yet another chance to respond in faith to light given. The condemnation of John 3 (quoted above) is not condemnation in the absence of light but in the active rejection, the hating, of light. This is why the call goes out that now is the day of salvation:

While it is said, To day if ye will hear his voice, harden not your hearts, as in the provocation. For some, when they had heard, did provoke: howbeit not all that came out of Egypt by Moses. But with whom was he grieved forty years? was it not with them that had sinned, whose carcases fell in the wilderness? And to whom sware he that they should not enter into his rest, but to them that believed not? So we see that they could not enter in because of unbelief. - Hebrews 3:15-19

The notion of a second chance after death for everyone turns the gospel message into something like, "Believe on the Lord Jesus Christ and you will be saved but you don't have to believe now because you'll get another chance later." Obviously this is entirely foreign to the entirety of the Biblical text.

Additionally, if there is a second chance after death wherein every person who ever lived gets another shot at believing in Jesus and, as some suppose, most or even all will believe because they will see His glory without veil, then this changes the biblical definitions both of faith and of salvation. Since salvation is clearly through faith, it falls upon those who hold to a second chance at salvation after death to demonstrate that faith is possible after death.

For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: - Ephesians 2:8

For we are saved by hope: but hope that is seen is not hope: for what a man seeth, why doth he yet hope for? But if we hope for that we see not, then do we with patience wait for it. - Romans 8:24-25

Now faith is the substance of things hoped for, the evidence of things not seen.

When Jesus returns in all His glory and every eye sees Him will this be saving faith according to the above definitions? Does saving faith wail?

Behold, he cometh with clouds; and every eye shall see him, and they also which pierced him: and all kindreds of the earth shall wail because of him. Even so, Amen. - Revelation 1:7

Generally, Bible literalists might believe in a second chance. It is just that none given the second chance will take it.

In Revelation 20 we are given a particular sequence of events after the glorious second coming of Christ. In the presence of everyone living, the dead in Christ are raised, then Christ returns in glorious fire. All the wicked perish in it. 1000 years later, all the wicked are resurrected too. However, it is a resurrection unto destruction; they are quickly destroyed again in the second death.

This is their second chance. They have witnessed the great power of God, however, refuse to worship His majesty. They are truly wicked in their hearts and perish a second time because of it. They were not raised in the first resurrection because they were wicked in their hearts; they would not have chosen Christ ever.

So as a matter of semantics, you may say there is this second chance, but for all practical purposes, because none given this 2nd chance will take it, there might as well not be one at all. Which leads to the purpose of the 2nd resurrection, but that would be another question, which has been asked from the traditional perspective.

This is a literalist interpretation of Revelation 20 which is mostly taken by the 7th Day Adventists and other Annihilationist groups.

Catholics don't believe in a second chance doctrine, as you phrased it.

But some do believe that Jesus can come to you at the very last second of your life. St. Faustina continually was visited by Jesus in dreams wherein she came to learn that God's greatest attribute it His Divine Mercy.

One thing alone is necessary: that the sinner set ajar the door of his heart, be it ever so little, to let in a ray of God's merciful grace, and then God will do the rest. But poor is the soul who has shut the door on God's mercy, even at the last hour.

St. Faustina's Diary 1507

Actually, Catholics don't believe in a first chance doctrine either. We leave nothing up to chance, you have the Life of Christ within you, and die, are judged and go to heaven. Or you go to hell or to purgatory.* If you make it to purgatory, you'll get to heaven eventually!

*Interesting synopsis of the Last Things by Bl. John Paul II

The mainstream Christian answer is that there is no chance for repentance after death. The Scripture used to support this is Hebrews 9:27 which says

And just as it is appointed for man to die once, and after that comes judgment

The protestant answer goes further saying this: (From http://www.gotquestions.org/second-chance-salvation.html)

While the idea of a second chance for salvation is appealing, the Bible is clear that death is the end of all chances. Hebrews 9:27 tells us that we die, and then face judgment. So, as long as a person is alive, he has a second, third, fourth, fifth, etc. chance to accept Christ and be saved (John 3:16; Romans 10:9-10; Acts 16:31). Once a person dies, there are no more chances. The idea of purgatory, a place where people go after death to pay for their sins, has no biblical basis, but is rather a tradition of the Roman Catholic Church

Now in all fairness, the Cathollic Church doesn't teach that there is a chance for repentance after death. This is a misrepresentation.

From the Catholic Answers forum:

The teaching of the Church is very clear on this point: if we die in a State of Grace, we will attain salvation. Otherwise, we (probably) go to hell.

We are placed in a State of Grace through Christian Baptism. We remove ourselves through mortal sin. We are restored to a State of Grace by Sacramental Confession.

The Church recognizes the possibility that salvation might be attained by other means (which have not been revealed). Thus, we have hope for salvation for unbaptized infants (but nobody knows for sure what happens to them)

There's no telling what we can make of near-death experiences. If we accept them at face value, they could be interpreted as an experience of purgatory (not hell). Or they might represent some "other means" of salvation that has not been revealed to the Church.

LDS teaching is a bit more nuanced. Some scriptures suggest that full exaltation in the highest degree of glory is impossible to attain for those who had an opportunity to choose to follow God in this life. Other scriptures add that full repentance is possible for those who had no opportunity to accept the gospel in this life.

LDS Scripture on those who had the opportunity to accept the gospel in this life:

Mosiah 2:36-39

And now, I say unto you, my brethren, that after ye have known and have been taught all these things, if ye should transgress and go contrary to that which has been spoken, that ye do withdraw yourselves from the Spirit of the Lord, that it may have no place in you to guide you in wisdom's paths that ye may be blessed, prospered, and preserved—I say unto you, that the man that doeth this, the same cometh out in open rebellion against God; therefore he listeth to obey the evil spirit, and becometh an enemy to all righteousness; therefore, the Lord has no place in him, for he dwelleth not in unholy temples. Therefore if that man repenteth not, and remaineth and dieth an enemy to God, the demands of divine justice do awaken his immortal soul to a lively sense of his own guilt, which doth cause him to shrink from the presence of the Lord, and doth fill his breast with guilt, and pain, and anguish, which is like an unquenchable fire, whose flame ascendeth up forever and ever.

Alma 34:32-35

32 For behold, this life is the time for men to prepare to meet God; yea, behold the day of this life is the day for men to perform their labors. 33 And now, as I said unto you before, as ye have had so many witnesses, therefore, I beseech of you that ye do not procrastinate the day of your repentance until the end; for after this day of life, which is given us to prepare for eternity, behold, if we do not improve our time while in this life, then cometh the night of darkness wherein there can be no labor performed. 34 Ye cannot say, when ye are brought to that awful crisis, that I will repent, that I will return to my God. Nay, ye cannot say this; for that same spirit which doth possess your bodies at the time that ye go out of this life, that same spirit will have power to possess your body in that eternal world. 35 For behold, if ye have procrastinated the day of your repentance even until death, behold, ye have become subjected to the spirit of the devil, and he doth seal you his; therefore, the Spirit of the Lord hath withdrawn from you, and hath no place in you, and the devil hath all power over you; and this is the final state of the wicked.

LDS teaching on those who died without a knowledge of the gospel...

Doctrine and Covenants 128:5

5 You may think this order of things to be very particular; but let me tell you that it is only to answer the will of God, by conforming to the ordinance and preparation that the Lord ordained and prepared before the foundation of the world, for the salvation of the dead who should die without a knowledge of the gospel.

All that aside, getting back to a purely Biblical answer, we fall back to the standard Protestant "No. Once you die there is no further chance for repentance, according tothe teachings of Jesus". For this, we need look no further than Luke 16:19-31

19 “There was a rich man who was dressed in purple and fine linen and lived in luxury every day. 20 At his gate was laid a beggar named Lazarus, covered with sores 21 and longing to eat what fell from the rich man’s table. Even the dogs came and licked his sores.

22 “The time came when the beggar died and the angels carried him to Abraham’s side. The rich man also died and was buried. 23 In Hades, where he was in torment, he looked up and saw Abraham far away, with Lazarus by his side. 24 So he called to him, ‘Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.’

25 “But Abraham replied, ‘Son, remember that in your lifetime you received your good things, while Lazarus received bad things, but now he is comforted here and you are in agony. 26 And besides all this, between us and you a great chasm has been set in place, so that those who want to go from here to you cannot, nor can anyone cross over from there to us.’

27 “He answered, ‘Then I beg you, father, send Lazarus to my family, 28 for I have five brothers. Let him warn them, so that they will not also come to this place of torment.’

29 “Abraham replied, ‘They have Moses and the Prophets; let them listen to them.’

30 “‘No, father Abraham,’ he said, ‘but if someone from the dead goes to them, they will repent.’

31 “He said to him, ‘If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead.’”

This horrifying account tells of someone who has died in his sin, and has no way out. Verse 25 makes it pretty clear that there is no way across that chasm from Hell to Heaven after death.