One of the little joys of math, well known, but always makes me smile: Though there are infinitely many rational numbers, they are countable.
I think 'infinite but countable' isn't so surprising—even without a formal definition, I think most people would expect the counting numbers to be countable—but maybe the fact that there are infinitely many more rational numbers than counting numbers, and yet there are exactly as many rational numbers as counting numbers?
(This is one of many ways of phrasing it, but it perhaps understates how much bigger the rationals seem to be than the counting numbers; the description I've given would apply as well to the set of all integers, whose countability is still perhaps surprising, but not as surprising.)
It isn't intuitively satisfying to say that there are "as many" rationals as integers because that is obviously not true; there are multiple rationals between any two integers. An argument to ignore that pattern as we scale up to infinity is rather flimsy.
But if we talk about infinite sets we are forced to admit the existence of well defined things that are plainly larger than |Z|, and no such well defined objects that exist between |Z| and |Q|.