  We study the iteration of transcendental self-maps of
$\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions
$f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential
singularities. We use approximation theory to construct functions in this class
with escaping Fatou components, both wandering domains and Baker domains, that
accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give
the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with
Baker domains and with wandering domains. In doing so, we developed a
sufficient condition for a function to have a simply connected escaping
wandering domain. Finally, we remark that our results also provide new examples
of entire functions with escaping Fatou components.
