Many exchanges don't require verification if you stay under the limit, Binance and Poloniex are 2 examples
But I'd still warn you about using these exchanges, even if they allow you to trade without verification, the fact that there is an option means they could still spring the surprise on you and ask to verify. In fact, I'm pretty sure in their Terms of service they will mention that verification can still be asked at their discretion.
Besides, as far as I know Poloniex will still ask for ID verification, I have forgotten it because I left this exchange for a long time because of some problems I got there. But there are still exchanges that only require email verification and usually I only use the exchange to trade a token, but to trade in famous coins which of course we will consider security more and verify id is not a problem I guess.