We study the escaping set of functions in the class B∗, that is, transcendental self-maps of ℂ∗ for which the set of singular values is contained in a compact annulus of ℂ∗ that separates zero from infinity. For functions in the class B∗, escaping points lie in their Julia set. If f is a composition of finite order transcendental self-maps of ℂ∗ (and hence, in the class 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 sequence e ∈ {0,∞}, we show that the escaping set of f contains a Cantor bouquet of curves that accumulate to the set {0,∞} according to e under iteration by f.