Here's a puzzle that starts with many solutions, but then I add restrictions and gets harder and probably still has many solutions:
Premise: Infinite supply piles, single player, you choose your initial deckstate and kingdom, but from then on you have the worst possible shuffle luck. Can you guarantee that you can gain a duchy every turn forever, following as many of the below restrictions as possible?
1. Without trashing any duchies, or revealing trader to the duchies.
2. Without setting aside the duchies.
3. Without ever drawing any cards apart from the cleanup phase.
4. Keeping the average cost of cards used below 4 coins. (This includes only cards that show up in your hand sometimes).
5. Without getting to choose your starting deck, only your starting hand.
6. Only using 4 different kingdom cards.
I can think of a solution that fits all 6 restrictions. Feel free to add more! The spoiler in #2 is because it gives a hint to how you might satisfy #1.
Watchtower, Travelling Fair, and Mandarin seem like the most likely culprits to me. Develop could be used, but it's really restrictive.
5 Poor House with Champion in play/+Action on Poor House gets you $20 for 5 Travelling Fairs, 5 Poor Houses(topdecked), and a Duchy. Not sure if having a Champion is part of "choosing your initial deckstate", but it only uses 2 different Kingdom cards(and an Event)
And, as eHalcyon said, 5 FG gets you $17 for Travelling Fair, Mandarin, and Province.
Also, FG, Copper, Silver, Plat, HoP for Mandarin buys Province with $9 and has $1 left over.
Diadem Silver Gold Contraband HoP is about as far as I'm willing to go with optimizing this, and gets you $10 for 2 guaranteed Duchies.