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GreenRoot comments on Rationality Lessons in the Game of Go - Less Wrong
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Go and chess provide clear demonstrations of opportunity cost, the first-mover advantage (esp. go), and the importance of not wasting time on trivial moves.
Go provides proof of, and some understanding of, the power of human intuition. My dad can make moves that I don't think he knows the reasons for, that turn out to have amazing consequences 10 moves later when I discover eg. that a group of stones of mine is dead because, even though I have more liberties in that group than he has in his attacking group, he can use his liberties while I can't use mine due to side-effects of those moves. But one is not inclined to view this as mystical intuition; it's patterns in the stones that his unconscious learned to recognize without his conscious mind knowing why.
Small advantages escalate. In chess, at the start of the game I might focus on trying to force my opponent to take a move back, or to trade a piece I haven't moved for a piece he has moved. Once that's done, I take advantage of my increased deployment to try to make an otherwise-even trade that disrupts his pawn structure. Once that's done, I try to take the isolated pawn. Once I'm a pawn ahead (I know then that I'll probably win) I force trades to make that advantage larger. I don't know if this works in real life.
Actually, one thing I enjoy about go is that small advantages don't escalate, at least not nearly as much as they do in chess. In go, if you make a mistake early that puts you behind by, say, 30-40 points, the place where you made that mistake usually interacts with the rest of the board little enough that you're not hugely disadvantaged elsewhere, and if you play better in the time and space that is left, you can catch up. But as you say about chess, I'm not sure if this is a very generalizable idea, at least when it comes to rationality.
For most practical situations I would suggest that it does generalise. Humans have relatively little ability to compound on success in a drastic manner. Exceptions of course include situations such as if Smily and Clippy were created at the same time on the same planet. Clippy getting the first week wrong could well leave tiling the universe with paperclips instead of molecular smiley faces is completely beyond his grasp.
It generalizes to real-time strategy games, at least.