Starting with the empty graph on p vertices, two players alternately
add edges until the graph has some property P. The last player to
move loses the *P-avoidance game* (there is also a
*P-achievement game*, where the last player to move wins). If P
is a property enjoyed by the complete graph on p vertices then one
player must win after a finite number of moves. Assuming both players
play optimally, the winner will depend only on p. All the games
solved in the literature have followed a certain pattern. Namely, for
sufficiently large p, the winner has been determined by the residue of
p mod 4. We say such a game has a *period* of 4 (or in some
cases a proper divisor of 4). In this paper we exhibit avoidance
games with arbitrarily large period. We also prove that the
equivalent games of 4-cycle achievement and 3-path avoidance have
period 7.