  We study the escaping set of functions in the class $\mathcal B^*$, that is,
holomorphic functions $f:\mathbb C^*\to\mathbb C^*$ for which both zero and
infinity are essential singularities, and the set of singular values of $f$ is
contained in a compact annulus of $\mathbb C^*$. For functions in the class
$\mathcal B^*$, escaping points lie in their Julia set. If $f$ is a composition
of finite order transcendental self-maps of $\mathbb C^*$ (and hence, in the
class $\mathcal B^*$), then we show that every escaping point of $f$ can be
connected to one of the essential singularities by a curve of points that
escape uniformly. Moreover, for every essential itinerary
$e\in\{0,\infty\}^\mathbb N$, we show that the escaping set of $f$ contains a
Cantor bouquet of curves that accumulate to $\{0,\infty\}$ according to $e$
under iteration by $f$.
