I found SouthXChange carries BNB and has no KYC. (https://main.southxchange.com). Not a giant exchange, but it's been around for a few years now and is reasonably safe to transact with in my estimation.
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BinanceUS really had no KYC to speak of for just getting BNB. Enter name, address, SSN, done! There is more required for higher levels (send PDFs of license/passport, bank statement), but none of that is necessary unless you're going to be depositing or withdrawing fiat.
There are several US States where Binance.US does not allow accounts, so for some people that's not a valid option.