You can trade almost most of the exchange without KYC, it's needed only for trading big amount. I have been trading in several exchanges without KYC and faced no problem till date.
However you can't trade on bittrex if you didn't do KYC. You will have 0 btc withdraw limit if you don't do KYC on there
Other top exchange like Binance, Kucoin allow users to withdraw without KYC and I think the amount is not that much lower as well. I have been using all the exchange without any KYC.