One could imagine defining various notions of higher-dimensional Catalan numbers, by generalizing objects they count. For example, because the Catalan numbers count the triangulations of convex polygons, one could count the tetrahedralizations of convex polyhedra, or more generally, triangulations of polytopes. Perhaps the number of triangulations of the $n$-cube are similar to the Catalan numbers? Or, because the Catalan numbers count the number of below-diagonal monotonic paths in an $n \times n$ grid, one could count the number of monotonic paths below the "diagonal hyperplane" in an $n \times \cdots \times n$ grid.
I would be interested in references to such generalizations, which surely have been considered—I am just not looking under the right terminology. I would be particularly grateful to learn of generalizations that share some of the Catalan number's ubiquity. Thanks for pointers!