We have indeed discussed this kind of thing in the past (e.g. allowing
users to earn "badges" for visiting a location, completing a tour, or
answering trivia).
A couple years back, there was quite a lot of buzz in the tech press about "gamification," but it seems to have died down a bit lately. For a fun reading of the possibilities, check out
"Reality is Broken" by Jane McGonigal.
It continues to be an intriguing idea for Curatescape projects, albeit one that comes with a lot of complications. For example, such a change would require a whole new interface, user authentication, a profile system with a social networking layer, and a whole host of other changes touching not just the iOS and Android apps, but also the front-end website design, and the Omeka backend. It's also not clear how well historical essays or other typical/existing Curatescape content would lend itself to gaming paradigms.
Game-based features probably won't be coming to Curatescape any time soon, but it's an idea we revisit pretty frequently when evaluating and planning development tasks so you never know.