How about the Leech lattice. This is a 24-dimensional packing of unit spheres where each one touches 196560 others. It is the densest 24-dimensional lattice packing (and very likely the densest 24-dimensional sphere packing, although this has not been proved). It has a remarkable amount of symmetry, and most of the densest sphere packings known in dimensions < 24 are derived from it (and known sphere packings in dimensions > 24 are nowhere near as dense when normalized for the dimension).
Maybe this is already implicitly included in the list, as it is closely related to the monster vertex algebra.
