I got a recommendation for hydrogen peroxide (use the stuff from the drugstore, not the higher concentration you could get at a healthfood store). The theory is that ingrown toenails are infections as well as nails growing in the wrong direction.
Dabbing my toenail with the peroxide helped some, but even frequent dabbing didn't help enough. Eventually, I put the peroxide on a small wad of toilet paper, and used my sock to hood it in place. And decided that I was going to continue with that for two weeks after all the pain was gone. Actually, a week was plenty-- my toe was starting to hurt (probably because I was using more peroxide), so I stopped, and my toe has been fine since. The skin was rough for about a week, but no actual damage.
Supporting evidence that there was an infection involved-- the last pain to go away was a little behind the center of my nail, nowhere near the corner where the nail had been digging in.
Edited to add: en_ki warns that leaving acid in contact with skin for extended periods can cause permanent damage, and "stop if it starts hurting" is not a good enough precaution. This should probably classified as cheap possibly safe rather than probably safe.
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