The functional properties of the entropy gives rise to 6 possible types of thermodynamics. Additivity or superadditivity or subadditivity are closely related to extensivity and this is one of the characteristics used to distinguish the 6 types. We give examples (some are somewhat academic) of all of these, except one. For this purpose we draw also on black hole systems which have been proposed. Some of these systems have subadditive entropies, i.e. they tend to fragment rather than clump. After proposing a new entropy function we raise the problem of how to select from these entropy functions. Are some better than others?