Hello, I have recently published Exploround app at:
It divides your neighborhood by square grid and remembers which tiles you already visited, awarding points for newly explored tiles and more points for randomly placed flags. The only competition is by comparing scores in various specialties on leaderboards - e.g. how far did you get, how densly you have covered the area or how much you explored without looking at map. There isn't any competition for the tiles, it's solely about getting to know your surroundings, finding hidden places and making your walks or runs more interesting. The app is free and the basic set of features will remain free.
If you'd like to try it out then any feedback is welcome.
Facebook contact: http://facebook.com/Exploround