That's the premise of the Ever Jane Kickstarter. Rather than going on quests to kill X number of monsters, players must worry about public appearances, lies, and gossip.
Currently, a prototype of the game is already available and free to download and play. "The current prototype provides fully functional infrastructure for both the gossip and the invitation systems as well as a 3D village in which you can walk about, bowing and curtsying to people appropriately," states the Kickstarter. "We have a tutorial that walks you through the basic mechanics and UI. There are no balls, dinner parties, or mini-games yet nor any ability to travel between villages. We need additional funds to build those."
The development team has already done plenty of work on the game. They're only asking for another $100K for development, a rather disconcertingly low amount for an MMO. To be fair, Ever Jane is targeting a niche audience that's probably willing to pay a subscription to play such an intriguing and different type of game.