I came up with this confusing theory a few years ago, but did not ask it on this site. If the Big Rip and the Multiverse theories are true, then when the Big Rip occurs, the universe's size will expand to infinity. However, the size of the Multiverse is also infinite (supposedly), and there are an infinite number of universes (also supposedly), then there is a paradox of infinities. I am still rather confused about this paradox, and is there a work-around to it, or is this completely invalid?
I don't know whether this is a real paradox. Infinities are tricky :) For example, we don't have just one infinity. Also, our knowledge about the universe is too narrow too understand the system of our universe (or multiverse) and we can only imagine the universes as some balls which are expanding into some other space = multiverse. If multiverse exists, it is probably not some space with some balls in it. There are a lot of dimensions that are present in here.
So, this isn't a paradox: we shouldn't think about the universes as some physical objects expanding. But we also don't know what to imagine. We just haven't figured it out. But it is probably a lot different from our current physics, math and imagination.
You have to take care of the kinds of size you are referring to, and sometimes also the type of infinity you get. This applies not just to multiverses but to a lot of other physical and mathematical objects.
The simple answer is that the "size" of the multiverse measures "how many" member universes there are, just as we can talk about the measure of a set of numbers. The physical size of each universe is different.
Consider a toy multiverse where there is a universe for each value of $w$ that determines the long-term expansion (they all start in the same state otherwise). The member universes with $w<-1$ will undergo a big rip at a $w$-dependent time. This multiverse correspond to an uncountable set, which is the same as the set of the real numbers. Had values of $w$ just been (say) rational numbers $p/q$ the toy multiverse would have been a countable set, in a sense a "smaller" multiverse despite still having infinite members.
Now, each member universe in this model has some spatial extent at any given time, unless it has undergone a Big Rip and the answer then is meaningless. The simplest case to imagine may be finite universes, like either a closed spherical geometry or some finite "hall of mirrors" geometry. Then we could assign a total spatial volume $V(t;w)$ to each universe with a given $w$ at time $t$. But we could have unbounded universes, where the volume is actually always infinite. This is fine, but requires talking about how the scale factor $a(t)$ moves things apart (it is what goes to infinity in a Big Rip event). The point is that whatever measure of how spatially big the universes are we use, it talks about something entirely different from the measure of universes.
It is like discussing the size of individual cars rather than the number of cars in the world. There is no paradox here.