I know that many exchanges allow you to fund your account, but when you try to withdraw they require KYC. I would like to avoid this problem, so I want to know which exchanges can be used without KYC.
I'd say that Binance right now would still be my go to exchange for anonymous, low volume trades. That may not remain the case, though.
Even though there are cases where they have asked for KYC seemingly out of the blue, I personally have never gotten that. There is always risk with dealing with exchanges that do not require KYC, despite what their official policy with it is as others say, because ultimately, legislation can change and that's out of their control.
The most important thing though is making sure that the funds are in custody of the exchange for a minimal amount of time. I'd definitely go with Binance, but you may also find other smaller exchanges without KYC requirements. But whatever exchange you use, make sure you don't use it as a wallet, because there are various issues that may arise especially if you aren't comfortable with verifying yourself.