We give a description of those functions f in the unit ball Open image in new window of H∞ on the disk \(\mathbb{D}\) whose orbit {f∘ϕn: n∈ℕ} is locally uniformly dense in Open image in new window for some sequence (ϕn) of selfmaps of \(\mathbb{D}\). An interpretation of this result in terms of the superposition (or substitution) operator on the space Open image in new window of holomorphic selfmaps of \(\mathbb{D}\) is given, too. Finally we present a new class of functions in H∞ whose orbit in H2 under the hyperbolic composition operator is non-minimal.