If you are willing to trade just crypto vs crypto then many exchanges don't require verification. Your ID check is a must if you want to exit to fiat
Honestly it's much harder to find anonymous exchanges these days. For instance bittrex used to allow people to have anonymous accounts but they no longer do. Only old users can remain semi anonymous, while all new ones have to verify. Same thing with poloniex, but they're taking it a step further by demanding verification from old users as well.
AFAIK you can still trade without verification on yobit.