Go

By all appearances, it's just two players taking turns laying stones on a 19×19 (or smaller) grid of intersections. But once its basic rules are understood, Go shows its staggering depth. One can see why many people say it's one of the most elegant brain-burning abstract games in history, with players trying to claim territory by walling off sections of the board and surrounding each other's stones. The game doesn't end until the board fills up, or, more often, when both players agree to end it, at which time whoever controls the most territory wins.