I'm guessing it's generally going to be 7. I was actually thinking about this a little earlier today. With 6 or less, you reshuffle after turn 1. That's not necessarily bad, but you're also likely to be reshuffling only a $2-4 card into your deck (the chance of hitting $5 is 1/21) before having two turns with that deck - assuming you get a Silver, your turns 2-3 are somewhere between a 6/2 (I believe a 1/6 chance), 5/3 (reasonably likely, can't be bothered to calculate it explicitly) or 4/4 (also likely, I think even moreso). Overall it means you're constantly shuffling a turn earlier but have a weaker deck each shuffle, which probably balances itself out. HOWEVER there's a good chance (I make it 5/42, or about 12%) you'll hit $2 on that first shuffle, which often just means you're permanently a turn behind. And I think that risk makes it slightly weaker.
Other values have a similar problem with shuffling, but start with even less buying power.
Could this be tested using Geronimoo's game-state simulator? Just setting one persons deck to 2-6 coppers and 3 estates, against 7 coppers 3 estates?