I dreamt this morning of a game where all six players have a small pile of legos. The first to play is chosen arbitrarily, and puts one piece in the center of the table. Turn passes to the player on his left, and each consecutive player must connect another piece to the growing creation. The goal of the game is to be the last player to make a valid connection.
We play a game like this with words: everyone adds another letter, but the goal there is to NOT complete a word when you add your letter.