I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the Fuchsian lattices may have as many cusps as they like as long as $\infty$ is one of them.
Does there exist a classification of the maximal elements of this set?
I don't think there's a simple description of these. For any cover $X\to E$ of surfaces (or orbifolds, if you allow Fuchsian groups with torsion?), there is a corresponding map between moduli spaces of $E$ to the moduli space of $X$. So I think to determine the maximal groups corresponding to a surface $\Sigma_{g,p}$ with genus $g$ and $p$ punctures, you would have to construct all possible (topological) covers of $\Sigma_{g,p}\to \Delta$, and look at the image of the moduli space of $\Delta$ in the moduli space of $\Sigma_{g,p}$. Then the maximal guys would be the ones that missed the images of these sub-moduli spaces. I believe in fact that the images will be subvarieties of the Deligne-Mumford compactification, so this space should be something like a quasi-projective variety (or something more general like a scheme or stack), but I don't know of surfaces in which this has been explicitly computed. However, I haven't done a literature search, but hopefully this might indicate how to make your question more precise, and search the relevant literature.
