I bookmarked this conversation over on reddit too late to join in the fun, but I thought that my Shakespeare Geek readers might get a serious kick out of what happens when you put the problem in front of geeks of the more traditional sense.
I can't really hold my own with the kind of mathematical experience they've got over there, but the way I've always imagined it is that "infinite" and "all" are, for the purposes of an abstract problem such as this, basically interchangeable. If you have a problem set of X possibilities, and then you say that you're generating an infinite number of variations on X, then by definition one of them will be X.
Any attempt to discuss how long this would take, or the odds that it could ever happen, or comparison to atoms in the universe, seems to miss the point entirely.
The closest I've seen to an argument that makes me curious is the idea that by saying "monkey" you are not necessarily saying "a true random number generator." Therefore you could argue that even with an infinite number of monkeys, your distribution does not follow a normal random distribution, and thus you can't do predictions based on that curve.