According to How to Create a Mind by Ray Kurzweil, the human neocortex contains around 300 million pattern processor modules, and "that a human master in a particular field has mastered about 100,000 chunks of knowledge." The primitive modules are assigned redundantly, with some things having many thosands of copies and others just a few. You can trade off overall capacity with integration/cross-corelation (e.g. the "fragmented hard drive" effect Jim mentions is predicted, and is more akin to reduced indexing). See also this answer for more background.
Experts already use the whole capacity of their brains. So an immortal won't continue to learn more and more, though maybe he will refine his ability in more subtle ways, optimizing his expertise and being able to try different methods by trial and error over the span of centuries.
The amount of storage for autobiographical information is also the same, so as he lives longer the amount he remembers will be a smaller and smaller proportion to the whole. But, so much is the same so what's lost, really? More interesting and unique experiences will continue to occur over time, so these will crowd out the mundane until that's all he ever remembers of the past. Talking to him, it would seem that he has had a very exciting life! But actually, the normal lifespan worth of excitement is spread out over thousands of years, with most of the time being mundane.
His own mind may perceive the past as exciting, while recent decades are boring. But it's always the case, as day to day memories fade except for the exciting parts.
December 2017: I came across a graphic story expressing a similar idea: Schlock Mercenary, Book 9: The Body Politic — Part II: Royal Flush.
As we grew older, less and less of short-term memory was deemed 'new' or 'different' enough to be committed to long-term storage. Entire years, and eventually decades would fly by unremembered. Sanity fled, and life became a cheap commodity again. Civilization itself began to die.