Have we discovered any natural satellites of natural satellites of planets or dwarf planets? Even very small, or relatively short-lived - e.g. ringlets around Saturn's moons, some meteorites orbiting Jupiter moons, or something to orbit Charon? Or is the Star-Planets-Moons the deepest naturally ocurring orbital recursion level?
Edit: originally I said "a moon with a moon would be an unstable system, due to the gravitational influence of the planet". @Florian disagrees with this. However, the answer is more complex than the Hill sphere alone.
At first approximation, the Hill sphere gives a radius in which orbits around a moon could be stable. Our Moon's Hill radius is 64000 km.
There is a prior answer here claiming that "a moon with a moon would be an unstable system". That is incorrect.
Intuitively: Of course satellites can have satellites with long-term stable orbits. Think of the Earth orbiting the Sun, and the Moon orbiting the Earth. The orbit of the Moon (a satellite's satellite) is long-term stable.
The orbit of a satellite's satellite will be stable if it's deep enough inside the Hill sphere, within the so-called true region of stability. The limits are a bit fuzzy, but the true region of stability is typically the lower 1/3 to 1/2 of the Hill sphere.
If you look at the gravitational potential, the Hill sphere is the area where contours become circular. Deep into that zone, orbits are long-term stable:
Bottom line is: A moon can have its own moons if it's big enough, and far enough from the planet, and if the secondary moons are close enough to the primary moon.
One way to calculate the Hill sphere is given on the wiki page linked above. Some more math can be found here:
A few extra articles about the issue of long-term stability of satellite orbits: