A series of books by Gene Doucette. The story revolves around a guy who is about 60,000 years old. He can't be exactly sure but that's his best guess. So, he's seen a lot of things, met a lot of people and been a lot of people. He's immortal but not invulnerable. About the only special defense mechanisms he has are an immunity to disease and rapid healing from wounds.
It's an unusual take on the immortality theme and Doucette's guy is very interesting. He's a raving alcoholic, for one. I've read the first three in the series; there are several more to go.