And that's totally, provably impossible, isn't it? /s
Depends why you want to stay anonymous. If you only want to hide the fact that you bought at a common sexshop or invested in a ponzi then an exchanges is alright. Though when someone is doing illegal things and the exchange can be forced by law to give details then this wouldnt be a good choice.
Though mixers would be still a risk too. There are not a few cases, in the US, where companies were forced to run their services while government is watching the data. They were not allowed to speak about that. So you never know. Trusting third parties... always a risk.