There are not a lot of options, some exchanges will allow you to trade till certian level is met and only then ask you for your KYC. And BTW Binance has high threshold till they ask for KYC. There are few anonymous "decentralized" exchanges out there, but they dont have the liquidity of the large exchanges.
It is true that the liquidity in these decentralized exchanges are not that great and with the regulations coming up we need to have strict actions being taken against exchanges who does not take care of the personal documents we are producing and if the regulatory body start giving out hefty fines like they did with facebook, every exchange will have a better security in taking care of their customer documents.
