  This article concerns the iteration of quasiregular mappings on
$\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are
always points at which the iterates of a quasiregular map tend to infinity at a
controlled rate. Moreover, an asymptotic rate of escape result is proved that
is new even for transcendental entire functions.
  Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be quasiregular of transcendental type.
Using novel methods of proof, we generalise results of Rippon and Stallard in
complex dynamics to show that the Julia set of $f$ contains points at which the
iterates $f^n$ tend to infinity arbitrarily slowly. We also prove that, for any
large $R$, there is a point $x$ with modulus approximately $R$ such that the
growth of $|f^n(x)|$ is asymptotic to the iterated maximum modulus $M^n(R,f)$.
