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How long can the chain of satellites of satellites be?
The Lunar Reconnaissance Orbiter is a satellite of the moon, which is a satellite of Earth, which is a satellite of the sun, which orbits around the center of the Milky Way.
That's a chain of 5. How long does it go?
The Lunar Reconnaissance Orbiter is a satellite of the moon, which is a satellite of Earth, which is a satellite of the sun, which orbits around the center of the Milky Way.
The answer depends in part on what you mean by "orbit", and the timeframe for what it means for one body to orbit another. If you allow objects that orbit for a short period of time, the yes, it is possible for moons to have moons. But if you want those orbits to have lasted for a good fraction of the current lifetime of the solar system, moons cannot have moons.
You picked the Lunar Reconnaissance Orbiter. The LRO doesn't quite count as orbiting the Moon. It needs to use fuel on occasion to maintain its orbit. The vehicle's fuel will run out in about seven year, and after that, the spacecraft will likely collide with the Moon in short order. What this means is that while stars can have planets and planets can have moons, that's about the extent at the lower end of the scale. No moons in the solar system have been observed to have submoons orbiting them.
Some have argued that moons can have moons, but the assumptions they make are a bit too spherical (as in spherical cows). Calculations of the Hill sphere implicitly assume spherical bodies. As bodies get smaller they tend to deviate more and more from spherical. For example, our Moon has seven large mass concentrations (mascons) on the near side and exhibits a marked discrepancy between the near and far sides with regard to crustal thickness. Arguing from the perspective of the Hill sphere also ignores perturbations caused the central star, perturbations caused by the planet's equatorial bulge, and weird perturbations such as the Kozai mechanism.
What about larger than a galaxy? The Milky Way is gravitationally bound to the Local Group, which in turn is gravitationally bound to the Virgo Supercluster, which in turn might be gravitationally bound to the Laniakea Supercluster. However, it's rather tough to call the Milky Way interactions with the Local Group an "orbit", and it's even tougher to label the interactions with even larger structures as "orbits".
This means that the chain you have found is about the extent of such chains. One has to view things with eyes squinted very tightly to claim longer chains.
Well, Proxima Centauri orbits the pair Alpha Centauri A and B with a period of about 550,000 years, and it has planets, likely including one Proxima Centauri C, at a distance of 1.5 AU. We don't know if Proxima C has a moon, but it absolutely could, which would make the chain one longer than the natural chain you mention:
Galactic Center --> Alpha Centauri A/B --> Proxima --> Proxima c --> Proxima c's moon
I would also question the notion that moons cannot have moons. The inner solar system is crowded, with lots of sources of perturbation. If there were an earth-sized planet out past pluto, such a planet could easily have a distant moon capable of retaining a smaller moonlet.
I can imagine a star like Proxima having a distant planet, which in turn has a somewhat distant moon, which has a moonlet. But I doubt it can go much farther than that.
The galaxy Milky Way may be in orbit around the center of a structure of many galaxies.
Each satellite should be much smaller than its center object. But the Moon is already too small, not all orbits around it are stable due to mascons of the Moon.
So the answer depends on how many larger structures are the central object for the Milky Way. We don't know if our visible universe is in orbit around a center.
