But they still would be limited to only what monkeys can actually do with typewriters given enough time or monkeys to do everything a monkey will do with a typewriter.

Not arguing this at all, I think a lot of people get hung up on this though because they don't actually know what's "possible or impossible" in our universe, which may not in fact have a good answer. All that aside, it's just a thought experiment to reveal the inherent problems with working with infinities, because the number of "possible" things that can happen are quite radical.

No, I agree that independence is necessary, not just because of “always”, but because if, as a crude example, your odds of hitting B halve each time you hit A, an infinite number of tries isn’t guaranteed to give you Shakespeare, even if the odds aren’t technically 0. My problem was that what you originally described wasn’t independence, it’s uniformity, which isn’t a prerequisite. And it’s up to 9 upvotes now so I don’t know what’s going on.

In a literal and real universe, if you had real magic and could organize this in reality without it collapsing into a singularity instantly, then not only would you indeed get Hamlet instantly, you will also get monkey poop sculptures in every possible configuration, including models of us typing these words on PC monitors made entirely of monkey feces. They would just be fecal scultpures because, as you said, a monkey can't poop actual electronics so pooping an entire, working internet with two people chatting is impossible.... probably. You would get typewriters thrown around randomly until some collapse into black holes, until some accidentally form whole, working machines that do a vast array of functions, until some mix with the poop sculptures and create models of whole worlds. If you want to limit it to a mathematical thought experiment, then yes you can make points about key distribution and so on, you can make limitations that will constrain the outcomes, but in the real world all we really have to create those limitations is the speed of light and other fundamental conservation forces, which would constrain contradictory outcomes in an infinite model.

Again, the idea is to understand that infinity is an absurd idea, and we can't work with infinities in any meaningful way.