The clique graph of a graph is the intersection graph of all its (maximal) cliques. A graph is said to be -divergent if the sequence of orders of its iterated clique graphs tends to infinity with , otherwise it is -convergent. -divergence is not known to be computable and there is even a graph on vertices whose -behavior is unknown. It has been shown that a regular Whitney triangulation of a closed surface is -divergent if and only if the Euler characteristic of the surface is non-negative. Following this remarkable result, we explore here the existence of -convergent and -divergent (Whitney) triangulations of compact surfaces and find out that they do exist in all cases except (perhaps) where previously existing conjectures apply: it was conjectured that there is no -divergent triangulation of the disk, and that there are no -convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle. Our results seem to suggest that the topology still determines the -behavior in these cases.