In LDS teaching, do gods and goddesses produce spiritual offspring forever?

There is actually very little LDS doctrine on the subject of the specific details of how Exaltation will work. (Most of the doctrine is devoted to the weightier matter of how to attain it in the first place.) But it is explicitly said that it will be a life of the same kind that God enjoys, so answers can be inferred by examining LDS cosmology on the current state of the universe.

From Doctrine and Covenants section 76:

22 And now, after the many testimonies which have been given of him, [Jesus Christ,] this is the testimony, last of all, which we give of him: That he lives!

23 For we saw him, even on the right hand of God; and we heard the voice bearing record that he is the Only Begotten of the Father—

24 That by him, and through him, and of him, the worlds are and were created, and the inhabitants thereof are begotten sons and daughters unto God.

Verse 24 specifically speaks of a multiplicity of worlds, each home to children of God as ours is. And again in verse 112, speaking of the destiny of people who have not lived as they should:

112 And they shall be servants of the Most High; but where God and Christ dwell they cannot come, worlds without end.

This concept is also mentioned in the Pearl of Great Price. From Moses chapter 1:

27 And it came to pass, as the voice was still speaking, Moses cast his eyes and beheld the earth, yea, even all of it; and there was not a particle of it which he did not behold, discerning it by the Spirit of God.

28 And he beheld also the inhabitants thereof, and there was not a soul which he beheld not; and he discerned them by the Spirit of God; and their numbers were great, even numberless as the sand upon the sea shore.

29 And he beheld many lands; and each land was called earth, and there were inhabitants on the face thereof.

30 And it came to pass that Moses called upon God, saying: Tell me, I pray thee, why these things are so, and by what thou madest them?

31 And behold, the glory of the Lord was upon Moses, so that Moses stood in the presence of God, and talked with him face to face. And the Lord God said unto Moses: For mine own purpose have I made these things. Here is wisdom and it remaineth in me.

32 And by the word of my power, have I created them, which is mine Only Begotten Son, who is full of grace and truth.

33 And worlds without number have I created; and I also created them for mine own purpose; and by the Son I created them, which is mine Only Begotten.

34 And the first man of all men have I called Adam, which is many.

35 But only an account of this earth, and the inhabitants thereof, give I unto you. For behold, there are many worlds that have passed away by the word of my power. And there are many that now stand, and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them.

36 And it came to pass that Moses spake unto the Lord, saying: Be merciful unto thy servant, O God, and tell me concerning this earth, and the inhabitants thereof, and also the heavens, and then thy servant will be content.

37 And the Lord God spake unto Moses, saying: The heavens, they are many, and they cannot be numbered unto man; but they are numbered unto me, for they are mine.

38 And as one earth shall pass away, and the heavens thereof even so shall another come; and there is no end to my works, neither to my words.

The important points here are that God does create multiple worlds, each of which is inhabited by his children, and that it is an ongoing process, that worlds reach the end of their existence and new ones are created, throughout eternity. It's reasonable to assume that the same will be true for those who reach exaltation.