The denizens of the alternate-history forum I have been a member of since high school include a fair number of fans of A Song of Ice and Fire in their number. And a world with as rich a history as Martin's has a lot of possible roads not taken, so it's no surprise that a lot of ASOIAF alternate timelines have been written and many of them have escaped from the members-only Alien Space Bat forum to places like fanfiction.net, which anybody can read.
Here's the newest one, "The King Nobody Wanted." It starts out with Robert dying of his wounds on the Trident soon after he kills Crown Prince Rhaegar Targaryen. Now the cause of House Baratheon and the rebellion against the mad King Aerys are in the hands of Robert's unappreciated little brother Stannis, currently trapped in Storm's End.
Interesting tidbits in the story include:
*Robert's Rebellion lasts longer due to the machinations of, among others, the Queen of Thorns. Randyll Tarly shows what he can do when he doesn't have the credit-stealing incompetent Mace Tyrell to deal with. It's still going on as of the last update.
*Lyanna Stark survives the fever that killed her in the "bed of blood" in canon and so Jon Snow has a mother. A mother who named him Rhaegar after his father...
*Point-of-view chapters from the perspective of Jon Arryn and Balon Greyjoy, as well as lesser-known characters like Garth Tyrell and Tytos Clegane (Gregor and Sandor's father--since this was written before A World of Ice and Fire, I'm not sure if that's his canon name) along with favorites like Ned Stark. We even get a sympathetic POV from the young Janos Slynt, who hasn't turned into the vile scumbag he was in canon.
*This hasn't been posted on fanfiction.net as of today, but Cersei Lannister's wedding night is a heck of a lot more pleasant for everybody involved. Cersei is a nasty piece of work, but given how much of a lout Robert is, I don't blame her for hating him.
*A surprisingly endearing portrait of the young Viserys Targaryen, before years in exile and poverty warp his already fragile (he is a Targaryen, after all) psyche.
So sit back and enjoy another "what might have been" in one of the bloodiest deconstructions of epic fantasy ever written...