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.
For our own Moon, we know that most low orbits are unstable due to mascons: mass concentrations below the surface which make the Moon's gravitational field noticeably uneven. There are only four inclinations where an object orbiting the Moon avoids all mascons and would be stable: 27º, 50º, 76º, and 86º.
High orbits above the Moon aren't all safe either: above 1200 km and inclinations of more than 39.6º, Earth's gravity disrupts the satellite's orbit. Note that these orbits are comfortably within the Moon's Hill sphere.
There are stable orbits at high inclinations and high eccentricity:
As for other moons in the solar system: most of them are smaller and orbit around larger planets, so their Hill spheres are small, and the planet's gravity will disrupt much of the volume inside the Hill sphere too.
Moons below the limit where their gravity is strong enough to make them spherical, will have problems with uneven gravitational fields. Mascons may also be present.
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:
It's theoretically possible for moons of moons to exist in stable orbits, gravitationally speaking. However, as far as I'm aware, no natural moons of moons have ever been observed as of yet. (Incidentally, there's a list of proposed terms for moons of moons, the most popular of which seem to be "submoon" and "moonmoon".)
There's a 2018 paper that explores this topic. Their calculations show that several moons in the solar system are actually theoretically capable of hosting long-lived moons of moons, including Saturn's Titan and Iapetus, Jupiter's Callisto, and Earth's Moon.
This article has a good summary of the paper.
It's true that the Hill sphere is an approximation, and that perturbations from other gravitational bodies and radiation can destabilize an orbit even within a body's Hill sphere. However, it's a decent estimate that if an orbit is within half of the Hill sphere's radius, then that orbit would be stable for on the order of billions of years.
The paper has some graphs showing that there are a few moons in the solar system that can have 10 km-scale submoons that are stable for at least the age of the solar system, under the influence of planet-moon-submoon tides:
The authors of the paper note however, that the above graphs do not take into account dynamic instabilities such as the Moon's unusual mass distributions, Sun-Earth perturbations, dynamical interactions between moons in multiple-moon systems, and dynamical scattering events between planets.
Iapetus's equatorial ridge might hint at the existence of a past submoon, though. Levison et al. (2011) theorized that this ridge is from a submoon-generating collision, where the submoon was tidally pushed outwards and the debris belt was tidally pushed inward to create the ridge. Alternatively, Dombard et al. (2012) theorized that the belt was caused by a submoon spirally inwards and being tidally shredded apart.
Why haven't we actually seen any submoons directly? One reason might be that they're too small to be seen. It'd be pretty difficult to spot something 10 metres wide orbiting the Moon, let alone Titan.
Though the fact that we haven't seen any of these theoretically possible larger ones suggests that there may be some other reason why they're not commonplace.
For example, it may be just too difficult for them to form in the first place, in the chaos of gas and dust circling a baby star. Tides also make the orbits of moons expand over time, so what is comfortable submoon real estate right now would not have been billions of years ago. For example, the Moon likely formed within several Earth radii from our planet, which would have been way too close to the planet for a submoon to be feasible. And perhaps it's just too rare and unlikely for a moon to capture an asteroid and have it become a submoon.
There is some speculation whether Saturn's moon Rhea has rings, and thus moons (components of the rings). There are indicators for it but they've never been observed directly. Of course it's astronomically possible that moons have their own satellites. But we don't have observed any as yet.