What kinds of data other than geographical are topologically spherical? I'm trying to think of a data set that is essentially topologically spherical. It's easier to think of cylindrical datasets (two dimensions, one periodic) or toroidal datasets (two dimensions, both periodic). 

Obvious candidates are geographical and astronomical, ground and sky; but I think the only thing spherical about the sky is its projection onto the ground, so it really just comes back to Earth.

I find it helpful to think about in terms of a circle's fundamental polygon:
Any data consisting of a three element vector of numbers  $v =\{x,y,z\}$ could be viewed as topologically spherical if, after converting to polar coordinates, there is only one datum for every value of the 3D angle $\theta$. Thus, every point in the data represents an exterior of a topological sphere. This is taking the term "topological" literally and formally.

There are also graph structures (acyclic) that have neighborhood structures that allows the graph to be mapped topologically to a sphere. For example, all graphs that can be mapped to Platonic Solids (i.e. vertexes to vertexes, edges to edges, 1:1) are topological spheres. This can include irregular neighborhood structures, too.

As far as practical examples of these, none come to mind immediately.