I think I could memorize an entire private key given the time and motivation. There are people who have memorized pi to thousands of digits, so why would it be impossible to remember a short string of numbers and letters?
It would not be impossible. However the stakes are much higher for you memorizing your private key (that potentially controls thousands of dollars worth of bitcoin) while someone memorizing the first thousand digits of pie for example would not lose anything if he made a mistake. IMO this added pressure will increase the chances of you forgetting some important part of your key
