Does LDS teaching hold to a multi-universe theory or an infinite universe theory in order to account for the doctrine of eternal progression?

Science estimates that there are only between 10^78 - 10^82 atoms in the observable universe.

The keyword here being "observable". The portion of the universe that is observable to us is merely a function of the speed of light and the amount of time since the big bang. The total size of the universe and the number of particles within is unknown. All current data suggests that the universe is flat (curvature of zero) which would likely indicate that it extends infinitely in all directions. This would in turn imply an infinite amount of particles from which an infinite number of beings could be composed of. However, it is also possible that it is positively curved, making it necessarily finite. In such a case, a multiverse would be necessary to accommodate LDS doctrine.

There is no definitive doctrine taught by the Church about reality being composed of a universe or a multiverse.

I note that Mark and Dougvj's answers contain a lot of educated speculation. That's fine! None of the proposed theories are inherently contradicted, as far as I can tell, by revealed doctrine. It is worth reaffirming, however, that they are just theories. As Elder Dallin H. Oaks (living Apostle in the Church of Jesus Christ of Latter-Day Saints) has pointed out, what is "known" about our postmortal experience is a lot less than we may imagine. He also offers a reminder that, while it is tempting to extrapolate from a smattering of quotes by Joseph Smith or Lorenzo Snow (if I had a nickel for every time someone told me what the "real" interpretation of "As Man is, God once was; as God is, Man may be" is, I'd be quite wealthy!), one church leader offering a statement once is not a pronouncement of doctrine:

"As to all of these, the wise cautions of Elders D. Todd Christofferson and Neil L. Andersen in earlier general conference messages are important to remember. Elder Christofferson taught: “It should be remembered that not every statement made by a Church leader, past or present, necessarily constitutes doctrine. It is commonly understood in the Church that a statement made by one leader on a single occasion often represents a personal, though well-considered, opinion, not meant to be official or binding for the whole Church.”

In the following conference, Elder Andersen taught this principle: “The doctrine is taught by all 15 members of the First Presidency and Quorum of the Twelve. It is not hidden in an obscure paragraph of one talk.” The family proclamation, signed by all 15 prophets, seers, and revelators, is a wonderful illustration of that principle."

_{Source: "Trust in the Lord" By President Dallin H. Oaks First Counselor in the First Presidency}

As Mason Wheeler has previously pointed out (According to LDS teaching, was Adam the first man created in all the worlds or just this one?), the Church doesn't tend to take doctrinal stances on instances like this because they are simply not relevant to us. To again paraphrase Elder Oaks' talk, the far more practical concern for us is whether we will get to heaven, than how its mechanics will work out. Personally, I agree with Mark's skepticism of the "we all become our own planet builders and universe-makers" theory, but I can see many easy counterarguments--just as we all grow up and "maintain sociality" here on Earth while running our own households, so might there be an inter-universal sociality of Gods. Or maybe not. Again, we really don't know. :)

This is a lot of words to say something that probably doesn't need as many as I devoted it, but it's been something on my mind recently, so this was an opportunity for me to articulate a lot of the things I've been mulling recently.

In short: The Church of Jesus Christ of Latter-Day Saints has no official doctrine on a "multiverse" or any alternatives. While members are not discouraged from pondering or theorizing, they are counseled to remember the extremely limited information upon which there is to base any such speculation.

I saw your query today as I was sitting through the Primary program in Sacrament meeting. Other LDS will understand this. 😇 I was searching Google hits on the query [mormon string theory] and your question was right there.

One of the problems with your calculations is the assumption that every man (and woman,incidentally) who achieves godhood goes on to create their own planet, etc. This is a common misconception among Mormons and non-Mormons alike. There is nothing in the LDS scriptures, which consist of the Bible, The Book of Mormon, the Doctrine and Covenants [D&C], and the Pearl of Great Price that compels such a doctrine. There are, however, indications that such a doctrine is false.

The Prophet Joseph Smith taught than the same sociality that exists among us here on earth exists in heaven. (D&C 130:4.) We don't go to heaven and then go off and do our own thing. We do things, like creating, together. LDS scripture supports such a conclusion.

The Book of Abraham in the Pearl of Great Price includes an account given of the creation of this earth that differs from the Hebrew Bible on one particular matter. The Hebrew Bible describing the creation tells that God created the heavens and the earth. (Gen. 1:1.) However, Abraham records that it was the Gods (plural), or Council of the Gods that was responsible for creating the heavens and the Earth. (Abraham 4:1.) Thus, the creation of the earth was accomplished by God AND all of the other gods of that celestial generation. Rather than having 1 million gods each creating innumerable planets, we would see 1 million gods cooperating in the same socialty that existed among us here, where we all work together to raise the neighbor's barn, creating innumerable planets.

As to whether or not such an LDS doctrine is at odds with the amount of matter observable in the universe, I doubt many LDS Christians would struggle to comprehend an incomprehensible god's power. Indeed, I suspect there is a similar dearth of non-LDS Christians struggling to understand the incomprehensible God. That, of course, is what faith is for.

Anyway, I appreciate the question because in pondering and composing this response a couple of answers to other questions I have had for many years became clear. But those questions and answers are unrelated to your query.