The best way to stay anonymous is to trade privately!
You need to find someone you don't know to trade in the real world with cash!
But there is no absolute concealment, only relative hiding!
I've seen offers that has been selling more than 20 bitcoins on a local casino here in my country. But such large amounts are very cautious and if you're about to deal with them, make sure that you are not alone.
Bring along your friends for your protection too and deal in a place where there's a lot of people, safety first. As said, OTC markets are the best way to purchase bitcoin if you want to hide your identity from exchanges. And another one is to deal with a reputable seller here in the forum.