Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?

I found online the claim (which we may as well accept for this purpose) that there are $32241$ words in Hamlet. Figuring $5$ characters and one space per word, this is $193446$ characters. If the character set is $60$ including capitals and punctuation, a random string of $193446$ characters has a chance of $1$ in $60^{193446}$ (roughly $1$ in $10^{344000}$) of being Hamlet. While very small, this is greater than zero. So if you try enough times, and infinity times is certainly enough, you will probably produce Hamlet. But don't hold your breath. It doesn't even take an infinite number of monkeys or an infinite number of tries. Only a product of $10^{344001}$ makes it very likely. True, this is a very large number, but most numbers are larger.

Some references (I am mildly surprised that no one has done this yet). This is called the infinite monkey theorem in the literature. It follows from the second Borel-Cantelli lemma and is related to Kolmogorov's zero-one law, which is the result that provides the intuition behind general statements like this. (The zero-one law tells you that the probability of getting Hamlet is either zero or one, but doesn't tell you which. This is usually the hard part of applying the zero-one law.) Since others have addressed the practical side, I am telling you what the mathematical idealization looks like.

my intuition says that time is countably infinite while the number of works the monkeys could produce is uncountably infinite.

This is a good idea! Unfortunately, the number of finite strings from a finite alphabet is countable. This is a good exercise and worth working out yourself.

Edit: also, regarding some ideas which have come up in the discussions on other answers, Jorge Luis Borges' short story The Library of Babel is an interesting read.

[Note that in my answer I am actually assuming that there are only a finite number of monkeys. I don't see what is gained by having both the number of monkeys and the time frame be infinite: mathematically speaking $\aleph_0 \times \aleph_0 = \aleph_0$, and it is somewhat confusing to contemplate infinitely many monkeys typing simultaneously: too much is happening at once. In fact, there might as well be only one monkey, or at any rate only one typewriter.]

Let me take the unusual (for me) step of considering the practical aspects of this question as well.

As Ross Millikan has explained, there is a simple mathematical model of monkey keyboard pounding under which it is easy to see that the claim is true: the probability that at least one of the monkeys will type out Hamlet approaches $1$ as the time $n$ approaches infinity.

However there is an assumption here: namely, that the pounding on the typewriter is random or sufficiently close to random. One way to formalize this is to say that after typing any $n$ characters, the probability of hitting any given key as the $n+1$st character is at least $P$, where $P$ is positive and independent of $n$.

The problem is that for actual typewriter banging, this is a very unlikely assumption. The issue is similar here to what happens if you ask someone to produce a random sequence of digits, say from $0$ to $9$, or even a random sequence of $H$'s and $T$'s (for "heads" and "tails"). Just closing your eyes and banging away will produce something very far from being random.

If the question is meant to apply to actual monkeys with their nonrandom motor behavior, then it is something else entirely. I would be tempted to say that the probability of producing Hamlet does not approach $1$ as time approaches infinity, but I'm not sure off the top of my head how to justify this.