In my mind, I'm comparing it to the Sun-Earth-Moon system. After all, the Earth is primarily a satellite of the Sun, but the Moon is still gravitationally bound to the Earth. Could something like this occur on a smaller scale? And are there any examples?
It's possible, but it seems like it'd be rare. The planets with the most moons are giant, and very far away from the sun. That means the moons will be very strongly bound to the planet and not get disturbed much by the sun. If our moon had a moon, it'd have to be just the right distance from it that it wouldn't collide with it (the moon's gravity is far from even due to it becoming tidally locked to the earth early in its history, and anything orbiting it would be somewhat unstable due to that). It couldn't be too far, either, or the Earth would destabilize it and it could get ejected from the Earth-Moon system, or impact the Earth or Moon.
I think a moon with a moon is just a rare situation which needs special circumstances. That a small, rocky body fairly close to the sun has a moon at all is probably a rare occurance, especially considering our moon didn't form along side the Earth. But obviously it's possible, since our moon is there. I suppose it's also possible for a gas giant to have a smaller gas giant for a moon, and for that moon gas giant to have its own moons.
FWIW, some asteroids have moons.
No "large bodies" that I know of. Certainly it is physically possible for something to orbit a moon; lots of spacecraft have been orbited around the Moon and other moons in the solar system.
As long as we're simply discussing hierarchies of orbits, the Sun orbits the galactic center and the Milky Way is gravitationally bound to the Local Group.
[EDIT: ...Well, there's always the Death Star. It is the size of a small moon, so I guess it would qualify as "large." ;) ]
[EDIT 2: Carson got me thinking about how you might actually find a meta-moon. The tidal forces are so extreme for a three-body system at planetary scale that I very doubt that a meta-moon could form in situ. But thinking about the comment that inner-planet moons are rare got me thinking about those moons, specifically Mars' moons. Phobos and Deimos didn't form with Mars; they are captured asteroids. I think that is by far your best bet for nature actually creating her own meta-moon.
But where? Asteroid capture is rare to begin with, even for full-size planets. For a moon to actually accomplish a capture without coughing it up to its parent planet, you want not just a large moon, but one that is relatively large compared to its planet- our very own Moon. It is much larger relative to its parent planet than any other moon of the eight majors, and one of the largest moons in the Solar System, period.
The other special case I see is Charon. It is so big, in fact, that Pluto and Charon are better thought of as a binary planet system, since their center of mass actually lies in the space between, not inside the larger body, like all other planet-moon systems in the Solar System. Nix and Hydra, therefore, also orbit the barycenter, at a greater distance than Charon, and with almost perfectly circular orbits. In that sense, Nix and Hydra kinda-sorta orbit Charon. That's the best I can do for you!
Saturn has two moons Epimetheus and Janus, which don't exactly orbit each other, but do interact stongly in a 1:1 mean motion resonance.
There is also the example of a 'trojan' moon, which doesn't exactly orbit around another moon, but is instead locked inside that moon's Lagrangian point, thus appearing to 'trail' the moon in a co-orbit.
The only known such moons are those in the Saturn system:
Saturn's moon Tethys has two Trojan moons (Telesto and Calypso), and Dione also has two Trojan moons (Helene and Polydeuces).