For many, the ultimate game of thought and strategy is Chess. However, I believe that the ancient Chinese game of Go (also known as Weiqi, baduk or Igo) is much superior. Although it seems simple at first, the level of strategy and thought is vastly more deep than chess, and the number of permutations on the board eclipses Chess by a boggling amount. For those who have not heard of it, check it out. It may change your life like it did mine!
I know how to play it but I'm really weak. It's nice that it's not yet possible to make strong enough engine to beat pro players. I mean lately they made such engine, but it can't beat pro players (from the top elo) every time. And as you may know the strongest chess engines eat pro players on breakfast.