The exchange Kucoin is a good enough exchange. If you don't want to be KYC'd, frankly, you have little options to use. LocalBitcoins used to be a place for anonymous transactions. The best way is still to use P2P fiat-to-btc transactions. The Mycelium wallet has a service for that.
https://mycelium.com/lt/help.htmlBinance 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.