The Monster Vertex Algebra (aka the Moonshine Module) categorifies Klein's $j$-invariant, in the sense that it is a graded vector space whose graded dimension is the $q$-expansion of $j-744$. More generally, vertex operator algebras often categorify modular functions and (quasi-)modular forms. This has something to do with invariance properties of torus partition functions.

The Monster Lie Algebra categorifies the Koike-Norton-Zagier $j$-function product identity, in the sense that the Weyl-Kac-Borcherds denominator formula of the Lie algebra is precisely this identity. More generally, physicists seem to use constructions with words like "BPS states" and "D-branes" in a way that categorifies automorphic forms on higher rank orthogonal groups (but I don't how it works).