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 <I>$P$-avoidance game</I> (there is
also a <I>$P$-achievement game</I>, 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 <I>period</I> 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.