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 carpet. (This appears in the Kapovich and Kleiner paper mentioned below, but must be standard---the point is the topological fact that any planar continuum with no local cut points is homeomorphic to a Sierpinski carpet.  Indeed, they prove a converse result, modulo the Cannon Conjecture.)

 - The boundary of a random group is homeomorphic to a Menger sponge ([Dahmani--Guirardel--Przytycki][1]).

 - Bowditch [proved][2] 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 ([Kapovich--Kleiner][3]).

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.


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?amp=&loc=refcit&refcit=1834498&vfpref=html&r=6&mx-pid=2755002
  [2]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1638764&loc=fromrevtext
  [3]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=kapovich&s5=kleiner&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=4&mx-pid=1834498