Although, I've often wondered how messy Mission gets without the Once Per Turn clause.
It's pretty horrific with Monument:
1. Build a deck that's guaranteed to generate $4 every turn (simply trashing your Estates/Shelters is sufficient if you don't add any stop cards other than your trasher and a single Monument) and contains at least one Monument.
2. Play Monument if it's in your hand; either way, play $4 and buy Mission.
3. Repeat #2 as many times as you can stand, gathering an unlimited number of VP tokens (remember, they're not component-limited).
What makes it worse is, unless there's some way to 3-pile, there's no way to actually end the game, meaning you have to eventually give a turn to your opponent. If he can mirror before you can end the game, it eventually comes down to a test of patience.
Alternatively, if you can gain a VP lead without sacrificing your ability to always hit $4, use a single copy of a Workshop variant to very slowly pile out Copper, Silver, and any Action card that provides +1 Action and either +1 Card or +$1 (either of which ensures you still hit $4).
You can also use the infinite Mission strategy to try to get the perfect hand (although situations where this works are probably rarer than might be initially thought; connecting Treasure Maps is the first thing that comes to mind, but that probably wouldn't be as broken or slow as other situations).
Basically, the ability to take infinite turns as long as you always hit $4 makes things go really slowly as long as you can accomplish at least something productive with at least some of those turns (even if it's just reducing the Copper pile by 1 once per shuffle). And that's assuming that people are playing to win legitimately; it gets even worse if someone just wants to stall until his opponent resigns.