When does one go for a double doctorate?
Generally you cannot get two PhDs in the same field. One could get a second PhD in a second field.

People who want to do research in multiple fields or in a multidisciplinary topic or wish to change fields may obtain multiple PhDs. Alternatively, as you suggest, people who want to remain students forever do that.

The main downside of doing multiple PhDs is that people may not see that you are moving on with your career. There is a career after getting your PhD, and obtaining a PhD is a small step along that career path. Hovering around getting multiple PhDs would be akin to getting multiple bachelor degrees. People may see you as a permanent student, not someone growing into a mature researcher.