Before jumping in with my ideas, I'd thought I'd mention Gulliver's Travels - there being immortal was something of a curse because you never die, but you did continue to age. You ended up a decrepit lump - very scary.
I side strongly with Anacky on the joy of being alive. There are many stories of long-lived "shepards" of families where an immortal keeps tabs and helps out. I'm also thinking "Requiem for Methuselah" from Star Trek. Imagine continuing your passion for painting or composing or writing. Though the distraction of procrastination might make it hard to actually get any of it done

So to pull it back more to Raygunray's original posting; have a strong bond with your family - they'll take care of you, and give you something to live for. Even if you remain "static", you would find purpose in that.
I have a story-in-progress (aren't they all) where immortality leads to two societies - Gears and Tanners. Essentially Gears (corruption of engineer) want to accomplish something with their lives and Tanners (from going to the beach and get a tan) are happy enough to live their lives in pleasurable pursuits.
Before the discussion gets into the nitty gritty, the definition of how you are immortal is important. Being aware at an early age will have a serious impact on how you think. Long term planning of elves only looks that way to us - same as a
Mayfly thinks we are long lived. And technically vampires are not immortal, they can die. Vampires just don't age, sort of. Some literature explains that they "purify" as time goes on and change, but I see them more like trees than true immortals.