Most of my player-generated fiction tends to be in more fiction-first games with game mechanics for players to introduce new fictional elements: e.g., Fate, Fabula Ultima, Cortex, etc.
In Fate a player can introduce new fiction into the game: i.e.,
Declare a Story Detail. There are guidelines and caveats to this. (1) The new fiction must (generally) be based on an Aspect in play and follow-the-fiction. (2) The player must spend a Fate point to invoke this Aspect. Your ability to perform this feat is (generally) limited by these two factors, as well as the consent of other players (GM included) at the table.
What does this look like in play? So here are several examples, including one that I know that I have used before.
Example 1: Disgraced Bodyguard of the Prince
The players are trying to sneak into the palace that belongs to the royal family to stop an assassination plot; however, they are being refused admission. One of the players happens to have "Disgraced Bodyguard of the Prince" as his High Concept Aspect. While roleplaying in-character, the player picks up a Fate point, and says as a former bodyguard of the prince, he knows a secret way into the palace. He says that there was secret passage from the kitchen that led to the gardens, and they can use that exit in the garden as their way in. Fate point accepted. Now this secret passage exists in the fiction. So here as a GM, I honor their "victory" by accepting that such a passage exists. However, there may be further complications related to that tunnel or possibly on the other side.
Example 2: The Magic Sword and the Blacksmith
In a separate game of Fate, the players need a magic sword to seal the prison of a dark evil. The players find the sword, but there's a complication: it's broken! (This was the result of a player's choice to
Succeed at a Cost with locating the sword.) So they now have the broken sword, but none of them know how to fix it. One of the players then has the idea of spending a Fate point to invoke one of her character's Aspect "Shady Sorcerer's Guild Connections." So the player figures that she may be able to pull some strings to find someone in the Sorcerer's Guild who could fix a magic sword, such as a magical blacksmith. I again honor their "victory" by having such a character exist. However, again, this does not make the problem auto-solved. Instead, when the players meet the magical blacksmith, she tells them that she can't fix it with what she has. She needs the proper equipment and materials to do the job, which she tells them. Now the players have a new quest.
I would like to reiterate that with both of these examples just because the players created new fiction in how they approached the problem does not mean that they auto-solved the problem. The first created a secret passage between the kitchen and the garden, but there may be complications down the line. The second created a new NPC. The moment they did so, that NPC became "mine" as the GM, and that NPC was unable to auto-solve the broken sword problem.
Of Apples to Oranges and Fate to D&D
The tendency in conversation at this point is to compare the relevancy of these aforementioned games with D&D is to dismiss it all as being all "apples and oranges." However, I don't necessarily think that is the case, and the reason is a man named Jonathan Tweet. If you asked the various designers of Fate or Cam Banks of Cortex where they, respectively, got the idea for Aspects and Distinctions, then they will tell you that it was a game called
Over the Edge by Jonathan Tweet, which used "concepts" for building characters. When we look at 13th Age, we can also see more freeform skills called "Backgrounds," which are similar to Aspects and Distinctions. All that's really missing from 13th Age are meta-currencies that can be spent to declare new fictional elements into the fiction. That's it. But would it really be that difficult to add them? Whether you like meta-currencies like Fate points, Plot points, or Fabula points is a separate matter from whether or not they could easily be slotted into games like D&D or 13th Age.