I don’t recommend it, but I’ve got a site with 807 categories. So you could just go for it an see how it goes. If you have a fast machine and enough ram, it might Just Work.
I think that my temptation would be to keep make the 100+ groups be categories and then use Tags for the Basecamp projects rather than making each of them be a subcategory. You can enforce category-level tagging, so you could make people posting in the Dogs category have to choose a tag when they post. It’ll be a bit of a PITA, but choosing among 30 categories is a bigger pain, I think.
And it may be that the group doesn’t really care if each topic is tightly connected to the former basecamp project anyway.