Hare & Hounds, also known as "The French Military Game" is a great step up from TicTacToe!

About French Military Game, from BoardGameGeek:

The French Military Game, sometimes called Hare & Hounds, is the smallest and most simple of all hunt games. It originated in 19th century France, and became popular with French military officers during the Franco-Prussian War of 1870-1871. An article by Martin Gardner in the journal Scientific American generated further interest in the game in 1963, and it has become popular with computer programmers due to the ease of implementing its simple rules. From the Rodent Roundup version: The chase is on! It's three crafty cats against one tricky mouse. Who will outsmart whom in this game of chase? One player controls the three cats. That player's job is to trap the mouse so it can't make a move (cats win). The other player controls the mouse. The mouse must try to scurry past the cats and get free (mouse wins). The cats may only move forward, sideways, and forward diagonally along the inscribed lines on the playing surface. Only one cat may move per turn. The mouse may move in any direction, including backwards, one move per turn. Similar to 1 contra 4 (4 gegen 4 und 4 gegen 1)