Is it an infinite number of monkeys, or is it infinitely-lived monkeys? If what you want is Shakespeare with probability 1 it matters. Because Hamlet is a fixed finite string of characters. That means the monkey has to stop typing when the string is complete. If we model the monkey as a process which every second taps a random key from the keyboard according to a fixed probability distribution, then to produce the Dithering Dane he must eventually repeat the space bar (or equivalently no key at all) until his terminal date.
If that terminal date is infinity, i.e. the monkey is given infinite time, then this event has probability zero. On the other hand, an infinite number of monkeys who each live long enough, but not infinitely long, will Exuent with probability 1 as desired.
(If your criterion is simply that the text of Hamlet appear somewhere in the output string, then a) you are sorely lacking in ambition and b) it no longer matters which version of infinity you have.)
Mortarboard Missive: Marginal Revolution.