Binance is still a good place to go if you aren't going to be withdrawing more than 2 BTC a day. That's usually what I use without having to send off verification. Bittrex is about to start even asking for a social security number soon. I really don't feel comfortable with giving them that. I can kind of understand the ID thing, but I think asking for social security numbers is going a bit too far. Try to avoid any of the places that make you give them this much personal information.
Yeah, as much as I hate Binance, their limit without kyc makes it very easy to access. But mind you, that doesn't mean they won't one day suddenly ask for kyc. my policy is always be prepared and keep only what you can absolutely afford to lose, be it on a casino or on an exchange. If they can kyc, they will, and you better be prepared to pass kyc.