TITLE: Inter-Kissing Number for Non-Spheres
QUESTION [5 upvotes]: In 3D, the maximum number of spheres which can inter-touch is $5$ (MO question Inter-Kissing Number for Spheres of Different Sizes).  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 one "official connection" (e.g., only $2$ of the $8$ cubes can be declared as touching at the corner).

REPLY [11 votes]: There is no upper bound. There can be arbitrarily many congruent convex solids which pairwise touch face-to-face. See Erickson, J. Kim, S. "Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes," for many references.
    
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