I am not aware of such an exchange these days. It became the norm for exchanges to request a lot of information from their customers to avoid the problem of money laundering and tax evasion. Crypto currencies became so mainstream in the last 18 month that exchange companies need to protect themselves. Being labeled as supporting criminal activities can be the end of an exchange.
If you want real privacy, have you thought about finding similar minded people in your area? You could either look for gettogethers of a crypto community, or maybe maybe some local adds that you want to buy crypto currencies. Doing transactions directly with each other - like paying cash for cryptos would give you the highest level of anonymity. Just make sure you meet somewhere in a public place and verify that the funds are real before bringing large sums of cash with you.