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.
Trading with fully KYC complied makes you safer, that means your name is registered as a verified trader and you can go after the site in the event your funds will be compromise, I hope everyone sees complying with KYC is not surrendering your privacy, but more on the security of your money.
Whether we like it or not, we will all be going to the stage where everything that we do in crypto using a 3rd party, will require us to pass the KYC.