There are a few exchanges that I like that have a $10K withdraw limit per day. Which is WAY above what I care about, but if you are doing multi BTC trades, it's worthless.
Yeah, there are two things that seem to determine whether an exchange will require KYC or not--one is the daily withdrawal amount, and the other is whether you're trading with fiat (and a third would be the exchange's regulatory compliance, I guess). I've never had to go through the KYC process on exchanges, as I don't trade much and never trade in large amounts. Sometimes my trades are for laughably low amounts.
And OP, I'm sure you got the answer you were looking for since all the options I know of were mentioned, but I just wanted to add that unless p2p exchanges become the norm, the probability is high that there aren't going to be any centralized exchanges that don't require KYC in a year or two from now. When Yobit closes up shop for good, you'll know that time has come.