We show that the fast escaping set A(f) of a transcendental entire function
f has a structure known as a spider's web whenever the maximum modulus of f
grows below a certain rate. We give examples of entire functions for which the
fast escaping set is not a spider's web which show that this growth rate is
best possible. By our earlier results, these are the first examples for which
the escaping set has a spider's web structure but the fast escaping set does
not. These results give new insight into a conjecture of Baker and a conjecture
of Eremenko