By the time children today are old enough to die from natural causes, we'll have a cure for that.
At some point I think we will 'cure' cellular senescence. Then humans can stay physically in their prime, indefinitely. But there are other limits, like memory. Because memories require physical space, there is only room for a limited amount.
I think we can get past that either naturally, by overwriting old memories, or artificially, by augmenting memory with technology. Research is already looking at brain augmentation with electronics.
So what will be the ultimate limit? Or will humans achieve immortality?
Obviously I can only guess, but based on my own experience of getting older, I tend to find myself living in the moment more. I would imagine anyone that can survive being in their own skin for 1000 years is probably going to be pretty zen. Probably not going to be very emotionally reactive. Not necessarily jaded, but cool in a "I've seen some shit, and this ain't shit" sort of way.
If someone is augmented, I assume memory recall, and extra processing power would be used. In this situation, its not a stretch to think people would all be connected. With enough processing power to sift through the "internet of minds," I think every person would trend toward the same personality. I mean, if 99.999% of your knowledge and experiences are the same as everyone else, genetic factors would be the only unique part of your personality. I think this scenario would mean the death of the ego. In a word: boring. Lots of assumptions in that conclusion though.
Psychological trauma would probably accumulate, and impair function. I don't think there's a real neurological problem with information storage, or types of thinking though. Because such a person would just forget things as they learned new things, they'd probably forget just about everything about their original self after enough time. That might look something like amnesia, just very high-functioning. Insanity and suicide would probably be more common that not in either case.
Which leads to me to wonder about mortality statistics. Assuming there are no backups/copies of yourself to respawn into, I wonder if accidents, murder, or suicide would be the new leading cause of death in that world?