Memory is conscious or unconscious. Apparently, every experience is recorded unconsciously. A finite brain cannot accumulate unconscious memories indefinitely. Would memory overload drive an immortal brain mad or would the brain merely stop recording? In Poul Anderson's WorldWithout Stars, the memories of immortals are artificially edited whereas, in Anderson's The Boat Of A Million Years, the handful of immortals learn within themselves how to cope with memory overload. Should we have been shown at least one who did not cope?
If an immortal being were to remain identical with his earlier self, then surely he would have to consciously remember earlier experiences some of the time? However, he would be able to remember any particular experience less and less often as he grew older. Thus, he would effectively become a different person, as if one had died and another had been born, but that is how life works in any case.