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.
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.
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...