Yes. Take a look at Instructure, the provider of the Canvas online course management system (disclosure - I was the Canvas admin for the college I worked for for 6 years, and I love the product).
Not only do they have a wonderful well documented REST API that actually works the way the docs say they do, the product is licensed under the AGPLv3.
The tricky part for it is that there are various hooks to services provided by other corporations. For example, the built in conferencing tool is actually Big Blue Button (also F/OSS) and that is hosted by Blindside Networks. There are also a lot of LTIs (LTI is an interoperability spec for course management systems and other tools to pass info back and forth - what user, course, send a grade back in, etc) that are 3rd party hosted, some integrated into the base product, some that the admin (or instructor, if allowed) has to activate/integrate, and some of which have license costs.
Never really got into the online communities (I prefer old fashioned listservs vs. online ticket/forum combinations) so I'm not aware of any institutions that self-host, and I'm not really aware of what the base AGPLv3 code base actually comes with compared to the paid-for-hosting version. Strangely, our hosting and support contract costs are just about what a license from Blackboard, D2L, etc. would cost :) This cost may have licenses for some of those 3rd party tools included, I'm not sure.