Universal functions for composition operators with non-automorphic symbol

For sequences (φ _n ) of eventually injective holomorphic self-maps of planar domains Ω, we present necessary and sufficient conditions for the existence of holomorphic functions f on whose orbits under the action of (φ _n ) are dense in H (Ω). It is deduced that finitely connected, but non-simply connected domains never admit such universal functions. On the other hand, if we allow arbitrary sequences of holomorphic self-maps (φ _n ), the situation changes dramatically.