Bittrex used to not care if you didn't provide any info, then they started freezing accounts randomly asking for ID scans, phones, addresses... yeah no thanks, im not going to give my dox to pontial scammers, I just want to trade some altcoins (and some people have their money frozen even after entering details)
I was using Poloniex, but similarly, my old account is now considered "Legacy" and is now a matter of time where they will freeze it and not leave me do anything unless I enter all kind of details which I consider unsafe. $2k withdraws a day anonymously used to be enough but now these times are going to be over soon.
What do we have? Livecoin and cryptopia are some exchanges im using to speculate with some altcoins but their volume is really low. Can you register in Binance and trade anonymously? Never tried to get in there.
I would look more closely into Binance, HitBTC and Kucoin.
You can trade at Binance anonymously. I have tried that and Binance only require a two factor authentication to start trading. But it's only have a 2 Bitcoin withdrawal per days.
You could, but remember every exchange logs your IP address, so to be really anonymous you have to use VPN or some other means to mask your IP.