What is the best noSQL backend to use for a mobile game? Users can make a lot of servers requests, it needs also to retrieve users' historical records (like app purchasing) and analytics of usage behavior.
Some factors you might consider:
Developer familiarity: go with whatever you or your developers are familiar with. Mongo, Couch, Riak, DynamoDB etc all have their strengths but all should do ok here, so rather than going for an unfamiliar solution that might be slightly better go for familiar and save a bunch of development time.
Ease of cloud deployment: for example, if you are using Amazon AWS, then DynamoDB is likely an excellent choice. Sure, you could use Mongo on AWS, but why bother? Other cloud providers have their own preferred db, for example if you are using Google AppEngine, it makes sense to use BigTable or Cloud Datastore.
Your use case seems both well suited to NoSQL and not very challenging since your data has a natural partition by user. I think you'd be technically ok with anything, which is why I'm mainly covering other factors.