There is, however, one element of chess that's unbalanced: the first-move advantage. At the highest level, the win rate for white (the starting side) is apparently a bit above 50%. It's an interesting thing to find in the world's biggest strategy game.
My brother pointed out that fine balance only really matters at very high levels of skill. In other words, unbalanced systems in games only matter if players know how to take advantage of them. It makes sense; I've often gone first in chess and I can't remember the last time I won a game. Along the same lines, tic-tac-toe only continues to exist because not everyone knows that the first player cannot lose if they know what to do and cannot win if the second player also knows what to do.
Because of this skill aspect, a game doesn't need to be perfectly balanced to be fun. In competitive Age of Empires II, it turns out that Mayan archers are the best units and have no effective counter once there are enough of them. Despite the existence of this 'exploit,' I've enjoyed lots of multiplayer Age of Empires because I play at a skill level where it doesn't really matter what nation I play as.
What's the perfectly balanced game? Probably Yahtzee, unless you know how to make the dice land in a certain way.