  Let $f$ be a transcendental entire function and let $A(f)$ denote the set of
points that escape to infinity `as fast as possible' under iteration. By
writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$,
we obtain a new understanding of the structure of this set. For example, we
show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$
and this leads to significant new results and considerable improvements to
existing results about $A(f)$. In particular, we study functions for which
$A(f)$, and each of its levels, has the structure of an `infinite spider's
web'. We show that there are many such functions and that they have a number of
strong dynamical properties. This new structure provides an unexpected
connection between a conjecture of Baker concerning the components of the Fatou
set and a conjecture of Eremenko concerning the components of the escaping set.
