Quote by gvjt
I understand how to calculate the capacitance of an isolated conductor, say a sphere, and I know that the associated formula involves the surface area of the conductor. Does all of this make the assumption that the surface in question is convex, or at least "nonconcave"?

Nope  you can make a capacitor out of any shaped materials  consider two biconcave lenses facing each other.
More specifically, what if the surface is extremely complex, like a fractal where the surface area might be extremely large,...

etc.
It will depend on the details of how the surface is arranged. You'll have to do the math.
Most complicated surfaces can be understood in terms of a network of regular capacitors.
Put another way, if I was to fashion an extremely convoluted surface, could I wind up with an extremely high capacitance for the volume the object is occupying?

Yes you can. The simplest example, I think, would be wrapping a regular parallel plate cap into a spiral.
Or perhaps here's a more concrete example: What if I fashioned an object consisting of hundreds of thin metal plates all connected electrically at their centers by a rod, but with an air gap between each.

A network of capacitors, in other words.
How do you have to join two capacitors together to increase the overall capacitance?
But I think you should be using, as your basic model of a capacitor, something involving
paired conductors ... one providing charge and the other receiving it. The concept of capacitance does not really make sense if there is only one conductor.