Leaving chess aside since I would put it into a category by itself, the best board game I have played is Diplomacy, by a very wide margin.

http://en.wikipedia.org/wiki/Diplomacy_(game)

No dice, no randomness or luck of any kind other than which nation you start with, just pure strategy and negotiation skills. This game was introduced to me by a high school math teacher who took over the chess club and slowly subverted it into the Diplomacy club. His only rules as far as negotiating with other players: no physical violence and no breaking into each others lockers.

The game’s one fatal flaw is that you need exactly 7 players to play it properly.