Compositionally universal meromorphic functions

For a sequence of holomorphic maps $(\vp_n)$ from a domain $\Omega_2$ to a domain $\Omega_1$, we consider meromorphic functions $f$ on $\Omega_1$ for which the sequence of compositions $(f \circ \vp_n)$ is dense in the space of all meromorphic functions on $\Omega_2$, endowed with the topology of spherically uniform convergence on compact subsets. We generalize and unify several known results about universal meromorphic functions and provide new examples of sequences of holomorphic maps, for which there exist universal meromorphic functions. We also consider meromorphic functions that have in some sense a maximally erratic boundary behavior in general domains $\Omega \subset \C, \Omega \neq \C$. As a corollary, we obtain that meromorphic functions on general domains are generically non-extendable.