1) The human brain can only record up to about 200 years or so worth of data, so, beyond that, it cannot really store memories, so people might live to be say, 500, but the continuity of the individual would disappear. (Frank Rosenblatt of the Rosenblatt perceptron estimated that 200 year number, by the way.) Living so long one cannot recall who one once was seems a dubious proposition at best.
2) Evolution works by successive generations. If people live so long that they do not have a need to reproduce very often, then interesting evolutionary consequences will arise that I can only dimly speculate on. Seems like while the humans are in a blissful state of ageless biological stasis, other species will still be evolving, say, dolphins evolving opposable thumbs, or the like, and ultimately, humans will become trapped in a time-warp, biologically speaking. They won't evolve, but other species will. They may be sentient (e.g. chimpanzees, dolphins, even highly evolved mollusks and the like perhaps), or, they may be non-sentient, like viruses and the like. If we stop evolving, we will eventually go extinct, simple as that. Oh, and, uh, sex, Yeah, if there is no more need to reproduce, we will lose our sex drive. We could still replicate ourselves via cloning, but that whole copulation bit, that will be over.
So, basically to sum up, I see an ageless society as vastly over-populated, in 'Soylent Green' proportions, with people not even having a sense of memories about who they are as individuals. People stop f*king, since there is no more evolutionary impetus for that, and eventually we are replaced by viruses, sentient mollusks, or both.
What the anti-aging people will have to come to realize is that only the gene is immortal, her carriers are not, and if they try to be, the gene will still win in the end.