My second-ever game with Dark Ages also had Rats & Market Square, which ended up being the majority opening, and it was pretty powerful. It works better if there is some support, but just a little bit goes a long ways. In particular, I don't think extra trashing is a necessity.
Basically, if you ignore the cantrips (Market Square), you end up with a deck composed entirely of Golds and dead cards (Rats). This is not a bad deck. It's not a great deck on its own, but it's better than it has any right to be, and you get it fast. Like, really fast. Your average hand will become 2-3 golds, and the rest dead cards. This is a decent-ish deck in its own right, but more importantly it is extremely fertile ground on which to build an amazing deck. You only need a handful of support cards, which are easy to pick up when you're usually pulling down $6+ with +buy.
Your goal is now to put together 3 Golds in one hand, and almost half your deck is gold. This is not a difficult task. If you just play like one Warehouse, or one Smithy, you should be able to hit that easily. With Adventurer (lol), it's a solid lock. You don't even need trashing, just card draw (or cycling). Trashing will help, but it's not strictly necessary. In the game I played, the only trashing was Altar, and I felt I got more mileage out of the bonus Duchies & Cultists than out of the fewer Rats.
Major caveat: The ratio of Gold to dead cards is extremely important. I was playing a 4-player game, where 3 players were doing this strategy, so we each got like 6-7 Rats before they piled, and a similar amount of gold. In a 2-player game, where you get 10 Rats to yourself, the Gold ratio drops from around one-half to around one-third and the economics start looking less awesome. This doesn't make the deck inviable, it just means it takes more support. You would probably need a trasher in that case.
An alternative would be to stick in some money-generating Cantrips. Then your goal is to generate $2 floating, and land 2 golds in hand from a deck whose non-cantrip content is half-gold. The floating $2 is probably the harder part of this task.
This is a lot of extrapolating from a small amount of experience, but I do feel like this strategy has legs.
