The obstacles for publishing something today are pretty much nil. Just put it on your website, or on the arXiv. If you put it on your website, make sure that you can prove later on what you put there when (https://en.wikipedia.org/wiki/Trusted_timestamping).
Self-publishing doesn't count as "publishing" for most purposes. Trying to claim academic priority based on something you uploaded to your website or the arXiv is a recipe for getting laughed at, or, even more likely, for getting ignored outright. At least outside of those very few fields where the arXiv has become somewhat established, like math, computer science, and to a lesser extent, physics.
You can (and should) always go for some more respected form of publishing afterwards, especially if you need points for academia. But if you were first, and you didn't document that properly, and you couldn't get it published in any "respectable" form, guess who is laughing now? If the arXiv is good enough for Perelman, it is good enough for me.
Of course, a certain amount of immunity to social values is helpful here.
One problem with this approach though is the "double-blind review" policy of some prestigious publications. If your stuff is on the arXiv, it's difficult to double-blind it. One of the reasons why I think double-blind is a bad idea.
arXiv isn’t a great cite. It’s like “technical report”, and it doesn’t really count for points so much as it is a courtesy provided to the reader (well, the only real points that should matter).
I once dug out the invention of IDE code completion coming from a structured programming system called Alice Pascal from the mid 80s via a magazine article. That was a fun investigation, but no one really cared.