It's a classical result of Ahlfors that, for any sufficiently nice n-connected domain $\Omega \subset \mathbb C$ there is a holomorphic branched covering $f: \Omega \rightarrow \mathbb D$ to the disk $\mathbb D$, which extends continuously to the boundary and maps the boundary curves of $\Omega$ monotonically onto the boundary of the disk.
Now, suppose we have $n$ positive integers, $d_1,...,d_n$, and denote the boundary curves of $\Omega$ by $\gamma_1,...\gamma_n$. Can one find a holomorphic map $f: \Omega \rightarrow \mathbb D$ which extends continuously to the boundary of $\Omega$ and maps each $\gamma_i$ onto $S^1$ with degree $d_i$?
Even the case of $n=2$ seems difficult--one can readily achieve this with a real analytic function $f$ by a pretty basic interpolation.
The proof of the Ahlfors result also seems to be less than helpful--it involves finding an extremal for the analytic capacity functional and then analyzing the regularity of the optimal function. See, for example:
Krantz's Geometric function theory: explorations in complex analysis, Theorem 4.5.9.