  We strengthen non-structure theorems for almost free Abelian groups by
studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality
lambda and a free Abelian group. A group is called epsilon-game-free if the
isomorphism player has a winning strategy in the game (of the described form)
of length epsilon in lambda. We prove for a large set of successor cardinals
lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality
lambda. We concentrate on successors of singular cardinals.
