The other day, I came across the description of a game on Wikipedia known as Cooper Young (not a very catchy name, I know). It is the essence of psychological games: it removes the concept of game mechanics altogether. It is now one of my favorite games.
It is played on a checkerboard of any size, or a simulated one on pencil and paper, which is how I usually play. It is a two-and-a-half player game (the game is for two players, but you need three people). Player 1 picks a row of the board, player 2 picks a column. Then players alternate placing pieces (or writing their initials) on the board until someone picks the intersection of the chosen row and column. The third person, the referee, is necessary to tell the players when the game has been won.
My next programming project is to write an AI for this game.