When Harry Potter discovered the Elder Wand and Gaunt Ring tucked neatly into his trunk a week after the final battle he chose to ignore them. He knew without a doubt that if he were to acknowledge the fact that they'd somehow found their way back to him, there would be research and eventually a grand adventure. He didn't feel like it, he was tired and ready to embrace normal. Later when it became clear he wasn't aging and didn't seem to be able to die Hermione would learn of this lapse and lecture him for a full half an hour without pausing.
By 25 he had died twice in the line of duty as part of the Auror Corps. Luckily Ron was his partner and helped to cover it up. Harry had also realized rather awkwardly that life with Ginny was not quite as appealing as he'd first thought, mainly because he kept catching himself checking out other blokes.
By 30 Hermione recommended it was time to retire from public life, people were starting to notice. Terms like well preserved were being used increasingly often in news clippings and with rather more skepticism than previously.
By 40 Harry had pretty much given up on solving the problem having spent the last two decades researching and finding nothing.
By 50 Hermione and Ron had become grandparents for the first time and were both successful in their chosen fields. Harry himself was doing quite well writing under an assumed name. He'd already produced four best sellers in the muggle world and one in the wizarding.
Hermione persevered; continuing to research Harry's seeming immortality far into her hundreds. Harry helped of course as did Ron but both men had already accepted what Hermione could not. Harry Potter age 145 still looked 17 and would likely remain that way until the end of the world.
Time passed, Harry's friends died as did their children. Soon no one alive remembered what Harry Potter had looked like without looking at a history book. He was able to reenter the wizarding world to purchase supplies and such with only the occasional remark about how much he looked like the young hero who had mysteriously disappeared years ago.
While Harry had expected to live to the end of the world, he had not really considered living past it. Long after Wizards had become all but extinct Harry lingered on so when the Earth had used up its resources and the Muggles headed out to settle other planets Harry went along with them. What else could he do? He was still quite wealthy all those best sellers continued to sell and new books were written.
So Harry would go from planet to planet, moon to moon until there was nothing left to see or do and then he would move on. People had stopped asking why he didn't age just assuming it was some technology they could not afford and all in all his life was comfortable.
Authors Note: This came to me after I spent all yesterday on the couch watching firefly with a terrible cold. I still have a terrible cold but this idea won't leave me alone. Let me know what you think.