How successful startups engage users and grow their user base by using game mechanics?

I'm looking for case studies of startups that properly used game mechanics to drive user engagement and growth of their user base.

If by "case studies" you mean formal, written analyses this answer probably won't be of much help. If you're just looking for examples, the mobile / location-based social networking space is proving to be an interesting proving ground for game tactics. In particular, compare & contrast:

• Google Latitude: no discernible game
• Foursquare: lightweight game where users earn badges & mayorships for checking in at places
• Gowalla: started out with a game where people could pick-up and drop virtual items at places. Later versions of the service don't seem to highlight this.
• MyTown: almost all gameplay, with little / no real-world value (IMO).

Newcomers to the space like Burbn are also adding elements of accruing points & leveling up, although it's unclear to what end.

This is a look at one space - I'm sure there are many other good examples, as well.