In 3D, the maximum number of spheres which can inter-touch is 5 (mathoverflow.net/questions/106120). This maximum reduces to 4 for unit spheres.
Is there a different shape (e.g., an egg, or a pyramid) for which these maximums are not 5 and 4? If so, what shape has the highest maximum? To avoid "corner touching" (e.g., 8 cubes could all touch at one corner), please additionally require that every "touch-point" have only 1 "official connection" (e.g., only 2 of the 8 cubes can be declared as touching at the corner).