1.) At infinity, probability disappears, since things are only probable if they can be otherwise at some time.
2.) One gets not just Hamlet or all the books in the British museum, but all of them backwards. Any signal will be buried in infinite noise. All possible information would not be usable or accessible. Sure, we would account for complexity, but how would would be able to locate it in anything outside of us?
3.) One doesn’t get all outcomes if they are tied to a time constraint: to get Hamlet fifteen minutes into one monkey typing is possible, but it does not become possible at infinity.
4.) It is a mistake to think that because we cannot be astonished to the point of incredulity at complexity arising from chance at infinity we therefore have a reason not to be so astonished at the claim that it arises in a finite time, or even in any finite time.
5.) We slip into a cosmological gambler’s fallacy when we think that a longer finite time (“billions and billions” or whatever) is more relevantly like the infinite than, say, twelve minutes. Infinite time does not make some possible outcome necessary because it is a really big number of trials but because it does away with any time for a possible outcome to exist, and so requires that it be actual. But this is exactly what can never be removed from a finite time.