Have you already acquired BTC and are interested in trading to profit from price fluctuations, or do you want to buy your first bitcoins?

If it's the former then you could try BTC-E or bitfinex, as they don't require you to submit your dox, stool samples, etc. if you're not planning on withdrawing or depositing fiat currencies.

If it's the latter then most exchanges will ask for the dox, but you can avoid this hassle if you decide to buy through localbitcoins.