Yes it is true that you can trade cryptocurrency using binance with your account not being verified but it is still recommended to verify your account if you are going to trade huge amount of coins and increase your daily limits. If you are looking for an exchange that doesn't require verification it is still depends on what you want to trade, I will recommend Binance and etherdelta for those erc20 tokens. Kucoin also does not require to have fully verified account for their users to be able to trade but I haven't tried trading on their platform so I can't say that it is safe to use it. Bittrex and poloniex used to be like this but now you will not be able to trade without your account being verified.

There are still lots of exchange who does not necessary need to verify your account to trade. Binance and kucoin are those two new promising exchange with high volume trade. Im using them both exchanger and i do not face any kind of problem using this two exchange.

Why you don't want to have a verified account in an exchangers? That's a good to verify your account in trading sir in case you encounter a problem you can get your account with proof

as someone who has been bitten by this issue and have some experience in the matter let me tell you that do not trust the exchanges that say they don't need verification. of course what they say is true but there is no guarantee that it remains true forever. they can change their rules over night like what bittrex did and block you out.