  We investigate the connectedness properties of the set $ I^{\!+\!}(f) $ of
points where the iterates of an entire function $ f $ are unbounded. In
particular, we show that $ I^{\!+\!}(f) $ is connected whenever iterates of the
minimum modulus of $ f $ tend to infinity. For a general transcendental entire
function $ f $, we show that $ I^{\!+\!}(f) \cup \lbrace \infty \rbrace $ is
always connected and that, if $ I^{\!+\!}(f) $ is disconnected, then it has
uncountably many components, infinitely many of which are unbounded.
