What is the Biblical evidence that there are ONLY three persons in the "Trinity"?

In the Book of Job, there are nine speaking roles. Job's wife has a one liner, so I will ignore her.

God and Satan speak in the prologue, God as Trinity (Elohim, the strange plural-singular construction, is used as his name) and Satan representing all his minions. Job and his wife also speak in the prologue.

In the main dialogue, there are six speakers: three friends (Eliphaz, Bildad and Zophar), Job, Elihu and the voice from the whirlwind.

Each person in the trinity is represented by a speaker:

• Job is God the Son, the Son of Man (he places his hope in the Son of Man in Job 16:21)
• Elihu is the Holy Spirit (Job 32:8)
• The voice from the whirlwind is God the Father (cares for his creatures)

Each of the three fiends plays an antagonistic role:

• Bildad is the anti-Christ (he insults the Son of Man in Job 25:5-6)
• Eliphaz is the evil spirit (he is visited by a fearful spirit in Job 4:12-17)
• Zophar is the Father of lies (he alone of the friends refers to God as Elohim in Job 20:29)

It is a peculiar asymmetry that Eliphaz and Bildad speak three times, but Zophar only once. Because of the assignments made, consider Zophar coupled with Satan. Then the sequence of Satan and the friends speaking is:

• Satan, Eliphaz, Bildad
• Zophar, Eliphaz, Bildad
• Zophar, Eliphaz, Bildad

Thus the sequence is Father of Lies, Evil Spirit, Anti-Christ.

Looking at the true persons of the trinity, the sequence is:

• Job (Son of Man), Elihu (Holy Spirit), voice of God (Father)

This pattern defines a chiasm (pyramid structure common in Hebrew literature):

A. Father of Lies B. Evil Spirit C. Anti-Christ C'. Son of Man (Christ) B'. Holy Spirit A'. God the Father

The assignments of the people to their roles in the Trinity thus has support in the details of their speeches as well as the overall structure of the document.

There are three persons in the Godhead and each has a speaking part in Job. There are three distinct antagonistic roles, and each attacks one person of the Trinity. If there were more persons, there would be more parts.

Covenant Theology

Redemption, as seen through the framework of Covenant Theology, requires at least two Persons. As Paul puts it:

For there is one God, and there is one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all, which is the testimony given at the proper time.—1^st Timothy 2:5-6 (ESV)

The Third Person is necessary to transmit the benefit of Christ's ransom to the world:

Great indeed, we confess, is the mystery of godliness:

He was manifested in the flesh,
    vindicated by the Spirit,
seen by angels,
proclaimed among the nations,
    believed on in the world,
taken up in glory.

—1^st Timothy 3:16 (ESV)

Jonathan Edwards' Economic Trinity

Jonathan Edwards theory of the Immanent Trinity characterizes the Spirit as the very love between the Father and the Son. He transferred this idea to the Economic Trinity. He writes:

So that in this verse [1^st Corinthians 1: 30] is shown our dependence on each person in the Trinity for all our good. We are dependent on Christ the Son of God, as he is our wisdom, righteousness, sanctification, and redemption. We are dependent on the Father, who has given us Christ, and made him to be these things to us. We are dependent on the Holy Ghost, for it is of him that we are in Christ Jesus; it is the Spirit of God that gives faith in him, whereby we receive him, and close with him.

And later:

The saints have both their spiritual excellency and blessedness by the gift of the Holy Ghost, and his dwelling in them. They are not only caused by the Holy Ghost, but are in him as their principle. The Holy Spirit becoming an inhabitant, is a vital principle in the soul. He, acting in, upon, and with the soul, becomes a fountain of true holiness and joy, as a spring is of water, by the exertion and diffusion of itself.

There are three roles¹ in the act of redemption that the Trinity fills:

1. The Father: The giver of benefit.

2. The Son: The purchaser of benefit.

3. The Spirit: The benefit purchased.

Edwards finds support for this construction (among other places) in Paul:

Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!

“For who has known the mind of the Lord,
    or who has been his counselor?”
“Or who has given a gift to him
    that he might be repaid?”

For from him and through him and to him² are all things. To him be glory forever. Amen. —Romans 11:33-36 (ESV)

Summary

We depend on there being three persons in the Trinity to have equal honor in the task of redemption:

And each person of the Trinity is equally glorified in this work: there is an absolute dependence of the creature on every one for all: all is of the Father, all through the Son, and all in the Holy Ghost. Thus God appears in the work of redemption as all in all. It is fit that he who is, and there is none else, should be the Alpha and Omega, the first and the last, the all and the only, in this work.—Jonathan Edwards, "God Glorified in Man's Dependence"

Footnotes:

1. I'm grateful to Richard M. Weber's paper "The Trinitarian Theology Of Jonathan Edwards: An Investigation Of Charges Against Its Orthodoxy" [PDF] for this way of expressing the roles.

2. Edwards translated the preposition in him. I've asked a question on whether Edwards' translation is supportable here.

Based off of this question:

• We know that there is only one God.
• We know that Jesus was the visible image of God in all of the Fathers fullness.
• We know that the Holy Spirit is also God.
• We know that we are all like Jesus, in that the Holy Spirit dwells within (some of) us.

Scripture makes no mention of any other part of the Godhead other than the four premises above. I cannot see how anybody would accept that we are any significant portion of that Godhead.

I would say that we do know there are only three persons in the trinity.

We are baptized into the name of 1) the Father, 2) the Son, and 3) the Holy Spirit.

God's revelation of Himself was progressive. After Genesis 1 was written, that was not the complete revelation of God to man. There was more to reveal. The "mystery" was revealed in Christ.

Long ago, at many times and in many ways, God spoke to our fathers by the prophets, but in these last days he has spoken to us by his Son, whom he appointed the heir of all things, through whom also he created the world. Hebrews 1:1-2

So, after God the Son Himself came to earth and revealed God to us, it would seem that this revelation is complete. What greater thing could God do to reveal Himself than to come to us Himself?

So, while the Bible does not explicitly say "Three in One", it does reveal God as such.