That's an interesting point. On the Discovery Channel show "Curiosity" in the episode "
Can you live forever?", Mythbuster Adam Savage hosted and pretended that he was a person who lived for ever. The show documents his journey from mortal to immortal using real science and plausible science fiction.
Near the end his brain basiclly got "full" and he could not store any more memories. Thus he used like a "cloud drive" to store is memories on external servers. That way he could save his core memories in his real brain and his axillary memories on a computer.
Which brings up another question about what would happen if he forgot those memories. Would he still be him or would it be if we just did not remember something? Likewise what if you gave fake memories or memories from other people? Total Recall anyone?
Some cool science fiction to think about.