There is the falacy of myth that someone immortal should have known every important figure of the eras they have lived through. Hancock pulled that stunt.
Nope, you live a quiet, ordinary life, and move on every 15-25 years. You may know a huge amount of stuff, but unless you’re getting a PhD every 20 years, you don’t have the history to be a CEO or Doctor or anything like that. Hence, unless you invested both wisely and carefully, you are never wealthy. Hell, even any wealth beyond modest would be inadvisable, as it only attracts attention and would take too much effort to move from personae to personae.
Likewise, no Wolverine/Highlander history of fighting in all the major wars. No way you’re going to risk being blown to pieces, permanently trapped under rubble and buried, or just plain discovered because your wounds tend to heal abnormally fast.
I would have thought one solution would be to simply move somewhere with a high transient population so hardly anyone notices you’ve been there for decades and haven’t aged much.
Did you read the thread? Or the OP? We know that’s the solution. The question is what’s the problem?
Let’s turn this around. Suppose you, the reader, stopped aging today. You look in the mirror tomorrow, and you don’t look any older. You don’t look any older the next day either. Or the next, or the next… At what point do you realize something’s up? At what point does it become creepy? At what point do you go to the doctor and ask what’s going on? At what point do you avoid doctors because you don’t want them to figure out what’s going on? At what point do you start trying to look older, say by dying your hair gray (or grey-er as the case may be)? When do you tell your family? When do you cut off contact with your family to go into hiding? When do you start to worry about the government and what do you do about it?
Of course I read the thread and the OP. There seemed to be fairly equal discussion of both the “What’s the problem?” and “What could Lazarus do about it?” going on. The latter was the one I most felt like addressing at the time, FWIW.
Those are indeed interesting questions, but I really think the answers depend on where you live. I mean, would anyone actually believe you if you said you were born in (say) 1879, no matter how old you looked? It’d probably be easier to change doctors and fudge your birthday every few years rather than try and convice people you really are 123. Especially if you don’t look a day over 35.
Poul Anderson wrote a novel, Boat of a Milion Years, with this exact premise. The immortality mutation is extremely rare, affecting maybe 10 - 20 people in all recorded history, and is not hereditary. One of the protagonists was born in Phoenicia in the 10th century BC.