Can a black hole undergo a Penrose process inside another black hole's ergosphere? I recently watched this video on black holes:
https://www.youtube.com/watch?v=ulCdoCfw-bY

And that statement should give you an idea of my level of understanding :)

I was wondering if a spinning black hole A could complete a Penrose process through another spinning black hole B's ergosphere. i.e. if it could travel through the ergosphere of another spinning black hole and emerge out of it the other side.  What would happen?  Perhaps this is related to the question of whether black holes can orbit each other (in two or three-body system)?  What happens if black holes orbit each other, but their ergospheres overlap?
Black holes can orbit each other (we know this from the gravitational waves from their eventual mergers) and there is no particular reason against a small black hole passing through the ergosphere of a large black hole. 

The main problem is avoiding a merger. The the two-black hole spacetime has no analytic solution and numeric simulations are messy, but a merger will definitely happen if the event horizons overlap. In any case, a rotating black hole has an ergosphere with an equatorial radius of $2GM/c^2$ and event horizon somewhere between this and the limiting radius $GM/c^2$. If we use the extremal case, then we can presumably just barely fit in nearly an identical mass and rotation black hole in the ergosphere. More likely a much smaller black hole could fit in. The direction of rotation would matter for the dynamics, since counter-rotating black holes would tend to coalesce (see this page for a plot of the relevant radii and where the innermost stable circular orbits are). 

A single black hole passing through the ergoregion would not be able to perform the Penrose process, but we can imagine a binary black hole pair that is ripped apart by tidal forces and then presumably the remaining hole would be ejected with a lot of energy. So, yes, we could get fast small black holes this way. 

In practice this system would lose a lot of energy through gravitational waves, making it likely that the holes would coalesce anyway. The exact limits require numerical relativity simulations to estimate, and this one would be a tough one.

(Note that the gravitational waves also carry away energy, up to 1/8 of the total mass of two equal-sized holes merging, but this energy is very hard to capture. Penrose did note in his classic paper that Misner was first in proposing black hole mining using a variant of this.)