Pretty sure that bitfinex doesn't require any, nor does poloniex. Of course pretty much every smaller exchange, such as liqui, yobit, and a whole lot of others don't require any ID as well. But you probably don't want to risk your money on these exchanges.

Quickest though is probably shapeshift, and i haven't seen anyone been asked for an ID even with thousands of dollars involved.

If you want to trade fiat without ID, use Localbitcoins.