Is it probable/possible? What would it look like? How would it effect the other bodies long term? How would culture change if intelligent life developed on the host planet?
It's possible, and has already happened in the past (the Apollo spacecraft and other lunar orbiters), albeit for a what period of time.
Depending on the relative sizes of the three bodies involved, it could be stable (if the moonlet orbiting the moon is small in comparison to the moon, and orbited close to it, it would most likely be stable), but as there isn't a general analytic solution to the three-body problem, there is no way to prove that it is stable. (You could approximate it by numerical analysis if you had a specific moonlet size and orbital radius in mind, but the error would increase over the simulation time).
As for how it might affect the culture of an intelligent species, they might come up with stories about the moon god/goddess and his/her companion/son/daughter/faithful pet, but would otherwise be similar in concept to any of the many cultural stories about the moon that exist on our Earth.
We human have sent satellites orbiting the moon. This shows it is possible, provided there is a significant different in masses.
At the very end the Moon orbits the Earth, the Earth orbits the Sun, the Sun orbits the center of the Milky Way, the Milky Way orbits something else...
Check out the Pluto system of Kuiper belt objects. Several objects orbit each other and a common center of mass in a chaotic system.