Oh, I found an exercise similar to this in Jackson's electrodynamics book. It's a series solution with an infinite number of image charges, so I suppose that's the best that can be done.
Suppose there are two conducting balls of the same radius and the balls are separated by some distance. If both balls are at the same nonzero potential and the potential at infinity is zero, does a closed form formula for the potential exist? Since I can't find the problem solved anywhere I...