There are only two known polyhedra (a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices) in which each face shares an edge with each other face. They are the Tetrahedron (4 faces) and the Szilassi polyhedron (seven faces).

The shape, which is torroidal (i.e. has one hole in it) was discovered in 1977 by the Hungarian mathematician Lajos Szilassi.

It's currently unknown if there are any other (non-convex) polyhedra with more than 7 faces which have the same properties.

For technical details see Regular toroids, Structural Topology, 13: 69–80, Szilassi, Lajos (1986).