The brain knows its job. It will forget everything except Your "CURRENTLY needed information", such as Your beloved ones, Your current address and the information needed for Your current job. At the time of the third PhD You will probabbly know nothing for the first two. Believe it or not, some people today, who have several high educations are experiencing the phenomenon of knowing too less for their first high educations
The good news are that they are alive, and they very well manage their currently needed information.
At the time the humanity runs out of space for storing information, it definately will be on another planet with more resources.
Assuming that we are integrating with technology, the evolution of the brain will change over generations. A more efficient way of problem-solving, for example, would be to focus on the macro and neglect the micro (as tech could deal with such problems). It's hard to predict how are brains will change over the coming millenia.
At present, this is all speculation, as we do not currently have a way to extend the memory storage of the human brain into a mechanical one.