Zeno's paradoxes of motion represent some of the earliest attempts to confront this beast, and although modern tools like calculus or the idea of an infinite sum seems to resolve the problem, I can't say that we've really solved it. It's more like we've brushed it aside and agreed to pretend it's been dealt with. Although I find myself on a small minority here, I still sympathize with Zeno.
One thought-experiment I've never been able to swallow, however, is Hilbert's Infinite Hotel. From the very beginning, it seems to want to assume premises that I would not want to grant... but this is not about me. It's about you, dear reader, and it's about exposing you to interesting ideas that will be worth your while to think about, so here's a short introduction:
And if you want more, check out the Grandfather paradox or the Chinese Room thought experiment.