Sometimes even just one piece of knowledge can turn you into a black hole. It’s why nobody remembers every digit of Graham’s (phone) number.

;)

his phone number is highly compressible though, it’s the decompression/serialization step that turns you into a black hole

I know we're all playing around here, but surely the answer must be that there's a limit rooted in the specific biological/chemical implementation of memory in the brain that our hypothetical "infinitely-long-lived, but forgetful person" would hit before the Bekenstein bound (probably long before) and therefore it's impossible to turn a brain into a black hole via that mechanism.