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.