I'm not famous and probably never will be, but I'd be lying to myself if I said the idea doesn't intrigue me. It's certainly a motivator...something I think has to do more with my E type than my mbti type. In this way, it seems, the systems clash...yet I can see how it works for me.
As for what means I'd pursue fame with (simply for the sake of fame), I do not know. I don't have any discernible talents...and really I'm quite a boring person once the initial glow wears off. In this way, I'm strange.
But, I find pursuing fame for fame's sake is an activity that is, overall, justifiable. People are motivated by different things. Money, Power, Love...and I've always just assumed those things didn't really matter to me at the end of the day...or rather, were achievable with the right amount of effort and appropriate strategy. Fame, I think, doesn't work like that...you don't have all the answers, and its achievability is something that isn't wholly dependent on you. Perhaps, rather unhealthily, pursing fame to prove to myself that my existence isn't completely futile is my main motivation. It enables you to define yourself however you wish to define yourself, but through others. If my existence is significant to other people, the masses, then who am I to say that I don't think my existence means all that much? It means something to other people...and it's a selfish person who wouldn't be moved in some way by that. These thoughts constantly plague me, and have quite honestly lead me to do things I regret. But, it's all a growing process, I think.
And like the op title says, there is a sort of lasting immortality that comes from being a celebrity. Fame (though perhaps still a shallow value to want) is no less shallow, I don't think, than any other value. It has intrinsic value.
Beyond happiness, I mean. Happiness being by definition the value wanted above all others.
I can't think of anything worth being famous about. It'd be cool to be recognized for being some pioneer or having a special set of skills, but I wouldn't know what those were, personally. Definitely not entertainment. I think it's kind of sad that most fame revolves around that (not that I don't like it. I'm a musician myself. It's just overblown). What would be really cool is being famous like Neil Armstrong or Christopher Columbus. Maybe 500 years in the future, someone will do it again. As for other people, I don't know anyone who talks about being famous. I'm sure I wouldn't hang around for long.