No limit. You can license them individually, or if you have over a certain number (10, I think?), you can apply for a kennel license, which then saves you money. But then you need to abide by the rules for kennels, including allowing inspections and such. The people (well, person) I know with more just licenses all her dogs individually to avoid the other regulations.
My condo association only allows two pets total, and I abide by it. No desire to own another pet right now anyway! I could see looking for a third when Meg is 100% retired and Gusto is at the level Meg is at now - but that's a lot of years down the line, and I don't want to live here forever anyway.