"What is known about the maximum number of facets of a single convex polyhedron that can tile in R^3 or R^n?"

For an infinite unbound polyhedron, infinite facets, formed by the Voronoi cells of equally spaced points on an infinite helix.

For finite polyhedra, it's believed to be 38 facets, the Engel-38 shape shown here found by P. Engel in 1980. Moritz Schmitt verified that for Voronoi cell space-fillers in a space group, 38 facets are maximal. These are plesiohedrons. However, if Voronoi cells aren't required, Engel found 3 other 38-sided stereohedrons that are very minor variations of the Engel-38 shape, pictured at the top of the page. Plesiohedra are a subset of stereohedrons. However, there is also the larger set of space-filling polyhedra, which includes the Schmitt–Conway–Danzer biprism. The set of aperiodic space-filling polyhedra is not well understood. According to On the Number of Facets..., the upper bound is 92, with special emphasis on SG(214). But no-one has beaten 38 facets.

For higher dimensions, A006227 says there are 4894 4D space groups. One could conceivably re-use Moritz Schmitt's systematic technique and compile Voronoi cells for all of them. However, easier would be to just pick the most blisteringly complicated of the 4894 4D groups and just analyze those. You can get an idea of how weird things get by looking at 4D dice.