After extensive testing with glasses of water, I’ve determined that it’s impossible to stably balance one infinitely thin circle on top of another. This is true whether the circles are the same size or different, and does not depend on the mass distribution of the circles. This is because two circles intersect at either zero or two points (unless they are the same, and that still isn’t a stable configuration).

Interestingly, two circles seem to be the *only* two dimensional simple closed curves for which this property holds; any other pair of curves can be scaled or rotated until they intersect at four points, and I believe these intersections can be arranged so that their convex hull contains the center of mass (for balance). I spent a while on the plane trying to prove this, but no luck yet.