I've used two Rats to threaten endgame with a reliable draw deck. In a nutshell, if you have an empty draw deck and two Rats in hand (or one in the draw deck and one in hand), you can pile out merely by trashing everything in play. Normally, this means all your points go poof, but if you play say Nobles first, this protects your points so you can gain the whole pile.
This also works well with Fortress (gain the whole pile in one go, nothing trashed), with Islands/Nv/Haven (hide the VP cards on mats), VP chips (you can pile whenever you are ahead), whenever your opponent has a negative score (e.g. -1 thanks to the curse you just gave him), or when you can buy enough VP at the end (e.g. you have a Chancellor in a 0/0 game).
This is best if you have a gainer in hand that can pick up the Rats during the action phase. Things like Iw/Iw or Tr/Armory work really well for out of the blue grabbing Rats (and also Islands, Great Halls, or cheap Cursers).
Now if you opponent is knowledgeable, they can counter, but sometimes you can keep them over a barrel a long time with just one or two Rats. When one opponent can 3 pile this turn, but the other, at best, can do so next turn you can completely through their game plan off. For the extreme example, take a Goons/Village/Council Room/Loan/Rats game. Say the opponent gets lucky and he gets ahead on Goons. You get $8 and buy two Rats. Now the opponent knows you can, on any turn, end the game and buy a bunch of copper for points. So knowing you can end the game before he gets another turn and score a lot of points, he has to take points this turn instead of building further. If you can score 27 points off 3 Goons (21 chips + province), he has to buy 6 cards this turn with his 5 Goons to prevent you from winning (assuming he doesn't dip into VP). That can be completely lethal to an engine.
Now this is very niche, but you can pile out the Rats really quickly. With cursers, looters, or other easy piles, this can end up being a 3 pile before the other guy's strategy even starts moving.
Other unrelated uses for untrashable Rats:
1. Turn Ironmonger into a more reliable village. This is pretty rare, but if there is a really powerful engine option, it might be able to work. Something like Ironmonger/Goons/Xroads/Library might be strong enough to warrant replacing coppers & estates with Rats, but not strong enough to skip the Rats.
2. Nerfing the opponent's Tributes. I doubt this would really work, but if the opponent is doing something like Kc -> Tribute, you might able to get rid of all your coppers & estates and turn all his Tributes into non-drawing villages instead of +coin or +card.
3. Possession defense. Again this seems unlikely to really work, but loading up your deck with Rats might well nuke his ultra-lean Possession deck. Something like: he has Golem/Possession x2 might make it viable force him to gain Rats or forgo using your deck for provinces. If he doesn't trash through junk to get Province buying hands, you then pile out some useless cantrip (like Pearl Diver). I suppose if you have the ability to top deck 5 Rats that might reliably counter Possession.
4. Rats might allow you to trigger some other on-trash benefit late game. Mostly I'm looking at Hunting Grounds - pick up a Rats so you can pile the Estates without using a buy (trash, gain, draw, trash, gain, draw, trash), though maybe there might be a case for Catacombs (e.g. use University to gain Rats/Catacombs, trash them, gain Gardens/Silk Roads). I definitely could see the odd tactical play for something like Feodum/Armory. You have 8 Feoda and 15 silver, top decking a Rats would let you bump next turn from 40 points to 42 (as opposed to the 41 from an estate gain) and perhaps allow you to do something like trash a Feodum & buy a duchy.
