The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi_1(X \backslash \Delta) \to S_d$ where $S_d$ is the permutation group on $d$ symbols. If this homomorphism is transitive, i.e. the image of $\phi$ acts transitively on $\{1, \ldots ,d\}$, then this data allows one to construct a unique connected Riemann surface $Y$ with a map $f: Y \to X$ which is a branched cover: when we restrict $f$ to $f^{-1}(X \backslash \Delta)$ it is a $d$-fold cover and around the branch points the monodromy is given by $\phi$. This is known as the Riemann existence theorem and it is proven by constructing the cover corresponding to kernel of $\phi$ and filling in the missing points with disks using $\phi$. If $X$ is compact, $Y$ will be as well.

My question is: given the Riemann sphere (or any other Riemann surface), which compact connected Riemann surfaces can one get if one is allowed to pick $\Delta$, $d$ and $\phi$ as above? This may seem like a trivial question: topologically any Riemann surface arises this way. But it is not clear to me what complex structures can arise. Alternatively, the question may be phrased as: is the map from such data to the disjoint union of moduli spaces of Riemann spaces of different genus surjective?

I would of course also be interested in literature discussing this or related questions.

Edit: the answers are correct that this question was easy. I was actually interested in the situation where we demand that $\phi$ assigns to a loop around a point in $\Delta$ permutation of consisting of disjoint cycles of length two, such that we get only branching of degree 2.

So I guess the updated question is: what restrictions can we place on $\phi$ such that we still get all compact connected Riemann surfaces this way?