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.

I think it really depends on what the person actually wants to do with the two different degrees. For example if someone has both a PhD in chemisty and biology to become a biochemist, their understanding of the subject would be much more versatile. It is absolutely not nessesary by any means, but it would just depend on the person and how many angles they want to approach a subject; some people prefer more than one way of looking at something.

Most universities in the US prohibit awarding of duplicate or comparable degrees. For instance; one might have earned a PhD in Management from Papua New Guinea University, and planning to earn another PhD in Management from Yale. In this particular case, Yale won't admit that student. Similarly, HBS, NYU Stern, Berkeley Haas MBA program FAQ sections clearly state that they won't accept students who already hold an MBA degree from another institution.

"University policy prohibits awarding of duplicate degrees. If you have an MBA or comparable degree from an institute of higher education, your application will be ineligible for consideration."

http://mba.haas.berkeley.edu/admissions/faq.html