After hundreds of in-game days exploring the black forest, sailing along the coast, making roads and bridges to avoid running in circles or missing a spot, I finally found him! It took exactly one year in Valheim. As expected, I couldn't believe it at first. I had so often been mislead by some random lights. I still don't really know if I was lucky to find it so fast, or unlucky to need that much time. The Elder was not on my (very large) starting island, and far enough from the center that there would be some plains closer than that. But I don't think that's particularly special.
To clarify a bit on the challenge, there is obviously no map, I also do not use exploits (troll mining copper was fine, but not trick to get iron). I learned at some point in this challenge that vegvisir stones can get you the direction of the boss even without the map, but I elected not to use that, keeping it hardcore. As a bonus, I also started in a random location instead of the middle of the map. In my case, it was possible to go from my random spawn to the center by foot (swimming a bit to cross large rivers). That was a big starting island. The random start honestly doesn't change much, apart from the very start of the game. It did take a while to find the center stone circle and Eikthyr, but that wasn't too bad, as I could use Yggdrasil as reference to look for the center.
I expected the Elder to be the hardest boss to find, considering how big the black forest is, and it's certainly been a crazy exploration. Now we'll see how hard it is to find the others.
For anyone tempted to try this challenge, I do recommend the random start. But you should probably use the vegvisir to at least get a sense of where to explore. The way I'm doing it takes a specific kind of crazy.
Source: https://www.reddit.com/r/valheim/comments/z3fgm1/day_365_in_nomap_challenge_i_found_the_elder/