On the face of it, there are many reasons to want to stick around. I am writing this sitting in a hotel in Paris, and one of the reasons which occurred to me first is that I could learn every language that has ever existed, or at least a hell of a lot of them. I mean, I did try Klingon when I was younger, but who has time for that these days? Is there a Rosetta Stone - Klingon? It would be fun to communicate with other nerds incognito. Then of course there's French, Spanish, and various dialects of English I haven't yet mastered.
Speaking of Klingons, one of the main reasons it would be great to live forever would be to see what the future is like. I read an article the other day which predicted the kinds of things we'll have in future, and I, for one, would love to be around when we finally perfect flying cars. And invisibility. And the holodeck (though I'm sure most people will use it for pornographic uses, rather than educational).
And speaking of tech, perhaps I could create some. I mean, I could study EVERYTHING. And you'd have to think, given enough time, I could become good enough at anything to be at least well-respected in the field. Perhaps I could cure cancer. Perhaps I could just live to see it cured. I would be an expert on quantum mechanics and the life cycle of the common house fly; I could invent the successor to the internet, or become the first man on Mars; I could form a world government, or abolish human trafficking worldwide. If I had enough time, what could I not achieve? Not to mention all the fun I could have. Adventures, and cultural experiences, and sex with strangers.
This is all assuming, of course, that my faculties hold. That my body doesn't simply continue to degrade to the point that I am a mere conscious husk of flesh. Unless you add this caveat, or something like it (perhaps a form of cloning, or invention of my consciousness uploaded to an android), infinite life begins to take on a horrible pallor. And that's not to mention the sheer drag of being alive sometimes. There are times even now when I cannot live with myself, when I get sick of the sound of my own thoughts bouncing around inside my skull, when I wish I could simply be elsewhere for a while. But how can one be elsewhere from oneself?
Even assuming vigour of mind and body, and a relative contentment with oneself, still problems arise. I say 'problems', but really I mean loved ones. They would need to be around, too. Imagine living on while your friends all passed away. Imagine watching your children grow old and die while you walked calmly through life unaffected. This, I think, would be the worst thing about being immortal, and surely it would be enough to drive anyone mad. Eventually, I think you would avoid making friends, for watching them blossom and decay in front of you would be too painful.
As hinted at above, I don't think the human mind is built for too long a term. We just don't have the capacity to go on forever, in the sense that it is inimical to our mental well-being. I don't mean that all older people are mentally ill, just that after a few hundred years, I think life would be an increasingly hard thing to deal with. Perhaps I am wrong, or wrong in certain cases, but certainly part of me knows and appreciates that I won't be around forever. The idea might be testable in a few hundred years, when technology and standards of living have lengthened our lives still further, but for now all we can do is speculate.
This leads me to the advantages of not living forever, apart from those, in a sense, described above. There are a lot of shitty things you won't have to do anymore, like ironing, and flossing, and being ill. You won't have to worry about work, or looking stupid in front of girls, or global warming, or cholesterol. And indeed, if I am correct in my supposition that being dead is exactly the same as not having been born (remember that? No, of course you don't), you won't have to worry about anything at all.
All of this is purely an intellectual exercise, since there is no way to offer anyone a real choice between death and immortality. At least, not at present. But the question remains: would you want to live forever? And it is an interesting question.