Programming

Right now it's no big deal to any AI company because more code means more training for the AI, but will we get to the point that they're happy with code output enough and then turn around claiming they own those?

At least in the US:

The vast majority of commenters agreed that existing law is adequate in this area and that material generated wholly by AI is not copyrightable.

So it seems unlikely that they would be able to claim any ownership.

As for the rest of your comment (the parts around ownership): you always own the copyright for any copyrightable work you create, including code. When you post on a website, according to the ToS of that site, you're licensing your comment/code/whatever to the website (you need to for them to be able to publish your work on their website).

Some (many, most depending on what you use) websites overlicense your work and use it for other purposes as well (like GitHub), but in the US the judges have basically ruled that AI companies can pirate whatever works they want without any attempt to license them and still be fine, so the "overlicense" bit is more of a formality at this point anyway.