I know you're saying you can't be a monkey on account of being a dragon, but I choose to interpret that as you saying, given infinite time, you couldn't write the entire collected works of Shakespeare because that's funnier.
Given infinite time, and the truest form of immortality, I could watch the universe fizzle out, collapse in on itself again, and expand again, where I could find a new early civilization, and reign over it as a god.
It means infinite possibilities within realistic parameters. Eventually, one monkey will hit all the right buttons in the right order because they're already hitting the buttons. However, infinite time does not mean that one of the monkeys will eventually shoot lasers out of its eyes.
Also, it's more than likely that the monkeys aren't using the typewriter for work, but rather there's a random object with buttons and they decide to play with it.