There are plenty of other possibilities. Here are a few examples:

 - The boundary of the fundamental group of an acylindrical hyperbolic 3-manifold with totally geodesic boundary is homeomorphic to a Sierpinski gasket (Kapovich and Kleiner).

 - The boundary of a random group is homeomorphic to a Menger sponge (Dahmani--Guirardel--Przytycki).

 - Bowditch proved that cyclic splittings of $\Gamma$ correspond to local cut points on the boundary.  In particular, the boundary of a graph of free groups with cyclic edge groups can be decomposed along cut pairs into Cantor sets.

The classification of 1-dimensional Polish spaces implies that this is a complete list of boundaries of 2-dimensional hyperbolic groups, in the sense that if the boundary is connected with no local cut point (ie the group has no splitting over a virtually cyclic subgroup) then it must be a Sierpinski gasket or Menger sponge.

I imagine that a classification in higher dimensions is completely out of reach, though I'm no expert.  (Misha Kapovich is active on MO and can provide an authoritative answer.)

I'll add full references tomorrow.