There's probable none such, but let's say we want to find like 3-4 that will not freeze randomly our account for verification (I read that kraken makes some random questions). I mean only depositing/withdrawing btc and trading usd/btc pair. Which would you suggest? From coinmarket can I see biggest liquidity:
Bitfinex Gemini GDAX Bitstamp Kraken BTC-E Gatecoin
Please also write if there is trading limit or how big is depositing/withdrawing limit.
All of the exchanges listed on there requires verification, except for BTC-e which randomly freezes accounts without any notice beforehand and has a really bad support team. So i don't think you're going to find your answer from that list.
Since you only want to trade USD/BTC then i suggest you try localbitcoins. I have actually never heard of localbitcoins freezing anybody's account, unverified or not. All the other exchanges are heavily regulated, and localbitcoins is your bet if you want to stay anonymous.
If you're exchanging small amounts, using BTC-e could work. But as i said they have support issues and if you were to use bank wire to deposit unless you trade p2p you're going to be asked for verification.