Bear with me here. Suppose I have a rotating black hole X. Then the region of space just outside its event horizon is rotating. Not just orbiting objects, but space itself is "frame dragged" by X. So tidal forces mean that if I orbit X just outside this radius, I'll experience a force trying to spin me. I mean, presumably the frame dragging has a derivative one way or the other, which will try to spin me, as well as the more obvious trying to stretch me.
Okay, so now let's move a small black hole Y into this region, give the tidal forces time to spin it up (it can't stretch), and then move Y away again (at this point, we're firmly in the realm of science fiction, but there's no physics being violated, just engineering).
Once we've spun Y up, it now evaporates faster due to the Penrose process. Rinse and repeat. Or leave Y there to evaporate. Whichever you prefer.
Let me be the first to point out that "in practice" (whatever that means in this realm) the increased radiation is vastly dwarfed by the energy that we put in which gets sucked into Y and makes it heavier. And my guess - but it's only a guess - is that you can't actually win here, even in theory. Some higher principle probably says that you have to make Y heavier by more than you gain from the spinning. But I've never seen such a principle. Basically, the aim is to make Y evaporate faster at the expense of X evaporating slower. Obviously entropy must increase, and that's the sum of the areas of the event horizons of X and Y combined, but so long as X gets bigger, I don't know of a fundamental principle which says that Y can't get smaller.
Please, someone disprove this nonsense :)
Edit: Maybe the extra spinning energy of Y makes it heavier by more than the amount that it loses through the Penrose process?