The image of the function $z^n$ is an $n$-fold covering of the Riemann sphere with two branch points at 0 and $\infty$.
Is there a holomorphic function that covers the Riemann sphere with exactly one branch point?
I guess more generally, I'm asking if there is some restriction on the number of branch points a holomorphic function could have.
I have no knowledge of algebraic topology and was wondering if this question had an answer within complex analysis.