Stuart, as far as the infinities go, I can imagine arguments that suggest that an infinite universe is more likely than a finite one, especially a finite one that is extremely large. For example, if the laws of physics were to turn out to be much simpler for an infinite universe, given our observations, that would be evidence in that direction. Conceptually, infinity is a simpler concept than particular very large numbers, so Occam's razor might lead us to choose infinity.

In fact I would argue that if your prior has a non-zero probability for infinite size, then there must be a size N such that once you have observed the universe to be bigger than N, you must conclude that the universe is more likely to be infinite than finite in size. For any such prior there must be such an N. So unless you are positive a priori that infinite universes are impossible, evidence could eventually convince you that it is likely.