In addition to the other excellent answers (from people who have infinitely more experience with the process than I), consider: It seems to be the case that for successful researchers, there is academic credit/prestige assigned to the PhD awarding department and advisor. If a person has multiple PhD's, then who gets that credit? At best this credit would seem to get split and diluted. At worst, one or the other is at risk of being forgotten entirely.

In addition to the other excellent answers (from people who have infinitely more experience with the process than I), consider: It seems to be the case that for successful researchers, there is unique academic credit/prestige assigned to the PhD awarding department and advisor. Examples: (1) Richard Lipton's blog. (2) Joel David Hamkins' blog. If a person has multiple PhD's, then who gets that credit? At best this credit would seem to get split and diluted. At worst, one or the other is at risk of being forgotten entirely.