I think that the Illuminator badge should be awarded multiple times. There is no reason why a few good (actually: subjective, broad) questions or answers to them should be worth multiple badges (Great Question, Famous Question and Stellar Question are all awarded multiple times), but continued efforts to improve the site in addition to large amounts of good answers isn't.
The Socratic Badge is awarded multiple times, and can be seen as the long term badges for questions. The Illuminator badge is the long term badge for answering and editing, so it would make sense that it would be awarded multiple times.
This would give the following users with more than one badge:
- At least 7 Illuminator badges: One user on SO. (Total, 2 extra badges)
- At least 5 Illuminator badges: Two users on SO. (Total, 2 extra badges)
- At least 4 Illuminator badges: Four users on SO. (Total, 4 extra badges)
- At least 3 Illuminator badges: Five users on SO. (Total, 5 extra badges)
- At least 2 Illuminator badges: Seven users on SO, two users on Mathematics, one on TeX, LaTeX and friends. (Total, 10 extra badges)
- At least 1 Illuminator badge: Forty-four users on SO, nine on Mathematics, four on TeX, LaTeX and friends, two on Unix & Linux, one on Super User, one on Science Fiction and Fantasy, one on Drupal answers.
I think that there shouldn't be much multi-awarded gold badges, but I think that Illuminator should be one of them, like Socratic.