The local structure theorem for non-constant holomorphic maps tells you that every point has a nbhd on which a given holomorphic function looks like (in some charts) $z\mapsto z^n$ for some unique $n$. $n$ is unique because you can read it of as the number of preimages. If $n=1$, the function is a local biholomorphism at that point. As is immediate from the nature of the map, the so-called branching points (for which $n>1$), are discrete.
so the set of all branching points is at most countable. in particular, for all but countably many values, the fibers have maximal possible cardinality, …