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This might be an American term, but auditing a course is simply taking one without registering and without intent of grade or credit.

https://www.merriam-webster.com/dictionary/audit Definition 2 under the verb form

I don’t know if it’s still true, but it at least used to be the case that one could ask a professor’s permission to just show up for lectures even without any formal relationship with the school.

I'm sure it's still true that you can ask! Whether it works I expect will depend on the professor.

AFAIK in many countries and universities around the world this is no problem in 90% of the cases.

Good professors are more than happy to have you there. One I even asked to take the exam, and get it corrected (to test if I really learned the content). And they allowed me to do that, I got the corrected exam at the end. Of course, it was not written in any book in the university.

Comes from latin "audire" to hear.

Not sure if english is your first language but you could audit language courses and get caught up.