Modal realism tells us there is an infinity of possible worlds, but how many are there exactly? Is it countable infinity ℵ₀, uncountable infinity 𝖈, or some other, bigger uncountable infinity?
Actually, there's an important objection to Lewis's Modal Realism along the lines of your question. This objection was developed originally in this paper by Forrest and Armstrong I think (don't have access to this paper from home). The idea of their objection is that given some principles Lewis is committed to, it would turn out that there simply couldn't be a set of all possible worlds---it would turn out that the "set" of possible worlds wouldn't have a well defined cardinality or something. If you're interested, I can try to dig up the technical details later.
Apparently Lewis wrote a reply in the AJP in 1986, but I don't know what he says there.
Swartz mentions a degenerate case of possible-worlds talk with only one possible world. A modal logic with zero possible worlds might be said to not have even our world as possible, as not even a world! But so two worlds might be styled the trivial case, with three the minimum nontrivial one.
Brunet mentions David Lewis' modal realism as having ℶ2-many worlds as a lower bound; the same article discusses other attempts to number the worlds. This paper, whoso wrote it I'm not sure, poses an upper bound problem in this connection, claiming that nothing less than absolutely many worlds are required.ℵ◊
Waiving that claimed need, then, we can simply define realist modal logics in terms of how many possible worlds they work with. If the number of them forms a set, yet then it seems like we would have a set of existence claims in play, such as are not established by "denying them is contradictory by itself," so there should be nothing amiss, by itself, with a postulate of, say, 16 worlds. So modal logics would have an "ensemble number" to their name, we might say. And so we can define such logics arbitrarily, in terms of arbitrary assignments of numbers of worlds to them. For example, generically refer to a modal logic with some special metalogical parameter-number k to its name, then say that k is the number of worlds for that logic ("is the number of the logic" = "is the logic of that number," so to speak).
ℵ◊Incidentally, that paper comes to the conclusion that it does in advocating for plural quantification, but Boolos himself was apparently content with something like truncating the universe of sets at the first fixed point of the aleph function (see Kanamori, pg. 78).
The Lowenheim-Skolem theorem says that any axiomatization that has any infinite model at all has at least one model of every potential cardinality.
So if your notion of a possible world includes the integers, and can be described in a finite language, there are more than any given infinite number of possible universes obeying its rules. Because if all of them were of a given cardinality or below, we could haul out the mechanism of Lowenheim-Skolem, and make one of the next cardinality up.
To see this another way, what could differ? Obviously, the value of some untracked constant could be different, and could take any value in a range of real numbers, so there are at least uncountably many variations. Now think about functions from the reals onto the reals. There are 2^c of those, and clearly some irrelevant one of those could vary any way it wants, and we would not notice. How about transformations of those functions into one-another -- there are 2^2^c of those...
Any set of consistent propositions describes a possible world.
To have infinitely many possible worlds would require infinitely many sets of consistent propositions.
Infinitely many sets of consistent propositions would require infinitely many propositions.
Are there infinitely many propositions? I don't know, but it seems widely accepted that no natural language can have infinitely many sentences.