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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 essentialsingularities. We use approximation theory to construct functions in this classwith escaping Fatou components, both wandering domains and Baker domains, thataccumulate to $\{0,\infty\}$ in any possible way under iteration. We also givethe first explicit examples of transcendental self-maps of $\mathbb{C}^*$ withBaker domains and with wandering domains. In doing so, we developed asufficient condition for a function to have a simply connected escapingwandering domain. Finally, we remark that our results also provide new examplesof entire functions with escaping Fatou components.

Escaping Fatou components of transcendental self-maps of the punctured plane