Q. What is the largest number of pairwise disjoint, homotopically inequivalent, simple closed curves that can be drawn on a surface $S$ of genus $g$?
Genus-$4$ surface. Image from a Mathematica StackExchange answer of @whuber.
For example, I think I see $9$ cycles on the above $g=4$ surface.
My question is surely answered in the literature, in which case a pointer would suffice.