They don't. No matter where you live, no matter what you do, no matter how low your profile, someone, somewhere will figure out who you are eventually. One person alone can never keep their identity secret forever. Which is why you get somebody else to keep it secret for you. Namely: governments. Now, I know what you're thinking. "Governments are who you want to hide your identity from!" Not necissarily right. See, people think of ET. They think shadowy men will come to your house in the dark of the night, take you away, and dissect you while you're conscious. To be fair, if you exist before the 70s or so, they probably will, but only under certain conditions. 687
Governments (theoretically speaking) don't dissect strange people solely because they are strange, but because they believe that by dissecting strange things they can gain access to beneficial information. The way to get the government to stop harassing you is to have something better to offer them than a chance at immortality. What about military leadership that has centuries of experience? What about a brilliant scientist who will never die? Replacing highly skilled dead people is a huge problem for upper level government function. An incredibly high skilled minister, secretary, or official who will never die would be a godsend. And few are better than the government at keeping secrets. Or, you could run down the private sector part of the skill-tree and pay the government not to dissect you. In any country without strict campaign finance laws, it isn't difficult to get the government to do what you want by making sure the people in it know they are there because you put them there and they can be replaced if you want them replaced. Either way, once you hit the late 90's, no government will be interested in doing much more than collecting a blood sample. If your immortal isn't willing to get his arm pricked or his cheek swabbed to give everyone else the same opportunity they have, maybe you've got a bit of a jerk on your hands. Although, that could make for an interesting character. 238