A Major body orbiting a star is a planet.
A major body orbiting a planet is a moon.
A major body orbiting a moon is a what?
Are there any known sub-satellites known to exist?
I have heard it referred to as a “Lunar Satellite”. (no I don’t have a cite, this is memory here) Rather unoriginal I know…but c’mon, let’s not burden the vernacular with more silly names. 
IANAA, but I think generally any body large enough to capture satellites is going to be sufficiently large to orbit a star and not another solid body (just trying not to use the word ‘planet’ here, but you know what I mean).
There’s also the factor that if a large object comes close to the satellite, it’s likely that the effects of the larger body will either crash it into something or simply set that object into an orbit around the main body.
Epimetheus and Janus sort of qualify for that. Here’s a simulation of their orbital relationship.
In general, three body systems (one in which the bodies can’t be mathematically reduced to individual couplets in terms of primary influences) are unstable. So a small moon mostl likely will not have a significant sphere of influence–where its gravitational field predominates over that of any other body–sufficient to capture a smaller object in a stable Keplerian orbit.
Stranger
Thank you for this post. I knew I was envisioning the problem right, but could not put words to it.
For a planet to capture a moon-- as Mars did with its two, as Jupiter did with its dozens – the asteroid-that-will-become-a-moon must be orbiting the Sun in an orbit that approaches the planet at a low enough closing speed that it will be warped into orbit around the planet by the dominance of the planet’s gravity over the Sun’s. For terrestrial planets, that figure is significantly low; for the gas giants, quite a bit higher. The surprising thing is not that Mercury and Venus do not have moons (to respond here to another thread); it’s that Earth and Mars do.
When the influencing bodies are not merely star (Sun) and planet, but star, planet, and moon, the likelihood of capture by the smallest, least massive of the three, i.e., the moon in this combination, become minimal. If you’ve ever seen one of those fantastic pictures of Io (or any of the Galilean moons) moving across the face of Jupiter, you know intuitively the point that’s being made here – with something that big nearby, the likelihood that something is going to achieve a stable orbit around Io (Callisto, Europa, Ganymede) by the accidents of orbital dynamics is pretty slender. I am not a spaceship pilot (neither a Major nor named Tom), but I’d venture to guess that intentionally bringing a spacecraft into orbit around a Galilean satellite is probably one of the more difficult problems to work, even with the ability to accelerate and decelerate the craft at appropriate times. For something in free fall to be captured by one would take, pardon the pun, astronomical odds.
Oddly, the Moon itself is the one most likely to be able to do so – its mass relative to its own primary and the star they both orbit is proportionately higher than anything else (excluding the Pluto/Charon doublet), and its “zone of capture” consequently broader.
My guess is that when we do learn the nitty-gritty details about extrasolar stellar systems, finding a “tertiary satellite” – one which orbits a moon – will be an extreme rarity if it happens at all. One possibility would be a brown dwarf orbiting a star, itself orbited at a distance by an Earth-sized satellite, which could then capture “space junk” satellites like Jupiter V-CCLXVII or whatever they’re up to in numbering them.
You’re forgetting about man-made satellites.
They can be timed and steered to go into any orbit.
The ETs living on the planet and its moon do not know where the object came from and as far as they can tell, it;s just another chunk of matter.
The asteroid Ida has a satellite. The article doesn’t say if there’s an actual technical term for such, though.
Ah! But without the timing, steering and orbit adjustment, my WAG is that such a system is unstable over time and the largest body (of 3) will tear the sub-satellite (smallest body) away from the moon (middle-sized body).
Which begs the questions: What have we orbited around our moon? Obviously, there have been orbiting components for the Apollo missions, but what else? Are there any satellites still orbiting the moon, after being sent up years or decades ago? How difficult is it to insert an object into stable orbit around the Moon?
There was Lunar Prospecter though it was due to be crashed into the moon at the end of its mission. The time line shows European, Soviet and Japanese orbiters as well.
Would a moon polar orbit that stays equidistant to Earth be stable? Would it stay that way or would it stay parallel to itself?
The Moon is inclined roughly 18° to 29° with respect to its rotational plane with the Earth, so a polar orbit wouldn’t be equidistant anyway. An orthogonally-oriented orbit would be metastable, in force balance between the Earth, the Moon, and the orbital momentum of the satellite; imagine a circular orbit that is slightly offset from a plane going through the center of the Moon and facing the Earth.
In reality, such an orbit would be easily perturbed by the existing Lunar orbital eccentricty, precession and nutation, variable gravitational forces from the Sun, et cetera. Even small inputs like the variation in gravitational attraction by Jupiter would be enough to quickly mangle the orbit. It would tend toward a more elliptical orbit that precesses toward the elliptic of the the Earth-Moon couplet.
Stranger