A monkey could type any infinite sequence of letters if it types at random. Since infinite sequences of single letters, repeating patterns, and those containing hamlet except one letter is wrong every time are all possible infinite sequences, it's possible that the money produces one of them.

Probability behaves strangely in infinite situations. A single monkey will almost surely produce the complete works of Shakespeare in infinite time... But this is partially a flaw of infinity in general.

As another example, let's say your monkey produces an infinite sequence containing hamlet. What is the probability of that particular sequence arising? It's 0. There is no chance of any particular sequence arising... And yet that one did arise! It was almost surely not going to be that one, but it was. The probability of any single infinite sequence arising is 0, but nonetheless one of them will be the outcome.