  Delay games are two-player games of infinite duration in which one player may
delay her moves to obtain a lookahead on her opponent's moves. We consider
delay games with winning conditions expressed in weak monadic second order
logic with the unbounding quantifier (WMSO+U), which is able to express
(un)boundedness properties. It is decidable whether the delaying player is able
to win such a game with bounded lookahead, i.e., if she only skips a finite
number of moves.
  However, bounded lookahead is not always sufficient: we present a game that
can be won with unbounded lookahead, but not with bounded lookahead. Then, we
consider WMSO+U delay games with unbounded lookahead and show that the exact
evolution of the lookahead is irrelevant: the winner is always the same, as
long as the initial lookahead is large enough and the lookahead tends to
infinity.
