On my alternate history forum, I came across an A Song of Ice and Fire timeline with the point of divergence being that King Robert Baratheon is killed in the final battle of the Greyjoy Rebellion. The assumption being that Robert, a glutton for fighting on top of food, booze, and sex, would be leading from the front despite being commander-in-chief of the entire war effort.
(Here's some fan-art depicting Robert plowing through some Ironborn with his gigantic war-hammer.)
I got involved relatively late in the discussion. Here's a rough sketch of the timeline that resulted. The board member Hero of Canton suggested that the point of divergence is that Balon Greyjoy and his brother Victarion lead an assault on Robert's siege engines in person. Robert kills Balon and Ned Stark kills Victarion, but Balon's personal troops go berserk at the fall of their lord and gang-pile Robert. Robert is wounded by a spear and the wound doesn't seem bad at first, but it festers and in a few days Robert is dead. Joffrey is now king, but he's only six years old. The dying Robert names his foster father (and current Hand of the King) Jon Arryn Regent and Lord Protector until his (supposed) son Joffrey comes of age, with his stern brother Stannis as the new Hand. He also releases Jaime Lannister from his vows as a kingsguard and names him the new Lord of the Iron Islands.
(Yes, he's a kingsguard and that's legally problematic. I'll get to that later.)
Fortunately for Balon's two surviving children Asha and Theon, Jaime is present to make up for his failure to protect the children of Prince Rhaegar by fighting off an attempt by the maddened Gregor Clegane to kill them. As a result, "the kingslayer" earns a second nickname for protecting the kids. Theon is fostered at Winterfell per canon, with Asha betrothed to Jaime to strengthen his position. Asha is entirely cool with this, since she views his battle with "The Mountain That Rides" as paying "the iron price" for her.
Unfortunately for most everyone, the nefarious Euron Greyjoy has escaped the Iron Islands, leading an exodus of Ironborn die-hards into exile. On his way east, he stops to raid King's Landing and carries off Cersei Lannister, Robert's widow who is pregnant with Myrcella at the time, while she was visiting the Sept of Baelor to pray for the health of her unborn child. His intent is to hold both mother and baby hostage and once the baby is born, threaten her to ensure Cersei's....cooperation.
Apparently there's a reference in the third book to the High Septon being able to release a kingsguard from their vows for a suitable donation, but the High Septon's price is higher than expected. Euron has used black magic in his assault on King's Landing and the Septon wants something resembling a Crusade to punish him. The "young hotheads" of the realm (Edmure Tully, Oberyn Martell, and Jaime himself) set off to rescue Cersei and take down Euron, with Tywin promising Cersei's hand to whomever rescues her.
Eventually Euron is brought to bay among the Stepstones between Westeros and Essos. In an epic three-on-one fight with "the young hotheads," Euron is apparently killed. Cersei and Myrcella are rescued, but there's a complication the board member AJNolte suggested--Cersei is pregnant with Euron's child. Although means of ending a pregnancy exist in this world ("moon tea" seems to function like a morning-after pill, with tansy being a full-blown abortifacent), the unfortunate Lysa Arryn shows what happens when it's administered later in the pregnancy. Ultimately Cersei bears Euron's bastard, who is given to Tyrion Lannister (he of the "cripples, bastards, and broken things") to look after.
And Tyrion will be able to raise him in a more appropriate fashion. In canon, Tristifer Botley was one of several boys fostered at Pyke after the Greyjoy Rebellion and while there, had something of an affair with the young Asha. Since Asha is promised to Jaime Lannister, he's not just going to be sent away when it's found out, but something far worse. Either the Botleys rescue him before he gets the chop or the whole house goes into exile in exchange for his life. The Botley lands are given to Tyrion, who marries into an Ironborn family that has historically not gotten along with the Botleys as a parting middle-finger to them.
(Meanwhile Cersei is married off to Edmure, since Oberyn would reject such a marriage out of hand and Edmure has a reputation for being soft-hearted. Hopefully this will be better for her than marriage to the abusive oaf Robert.)
However, it turns out Euron Greyjoy isn't actually dead. His sorcery saved his life and per AJNolte's suggestion, he seeks out those who hates the Baratheon regime as much as he does--the exiled Viserys and Danaerys Targaryen. He takes Viserys as his evil apprentice, while Danaerys ends up married off to Khal Drogo per canon.
Meanwhile, back in Westeros, Jon Arryn bring the rising star Petyr "Littlefinger" Baelish to court and per canon makes him Master of Coin (this was my idea). Stannis, however, does not particularly like a man who runs brothels in King's Landing, and keeps a very close eye on how he spends and invests the king's money. Robert has been king for several years before dying at Pyke, so I imagine the realm's finances aren't in the best shape.
Ultimately, things come to a head. Littlefinger provides funding to Euron Greyjoy with the goal of Stannis leaving King's Landing to fight him. This gets him out of Littlefinger's hair for the moment and will hopefully get him killed, with other potential Hands like Mace Tyrell less likely to keep too close an eye on him. Meanwhile, Varys the closet Targaryen loyalist is still there, providing assistance and information to Viserys and Danaerys and "Aegon." Joffrey will have a better upbringing than in canon but six-odd years of being ignored by his "father" and spoiled by his mother during a crucial period in his development and whatever inborn psychological issues resulting from being conceived of incest are still going to be there. He's going to try to assert himself as the king more and more, with Varys no doubt trying to enable his worst aspects to pave the way for a Targaryen restoration.
If war comes, I'm imagining it as follows. Euron kills Stannis in battle and raids the eastern coastline with his understudy Viserys while Khal Drogo and "Aegon" and the Golden Company begin preparing an invasion of Westeros per canon. Varys poisons Jon to sow even more chaos, while Doran Martell waits to unleash Dorne's army. Meanwhile, a grown-up Tristifer leads a fleet to the Iron Islands to get back with Asha over Jaime Lannister's dead body.
(In the books Asha thinks Tristifer thinks she's some damsel in need of rescuing and in this timeline you could make a case she is--her father has been overthrown and she's being married off to one of the men who did it. Tristifer can still be the dorky and well-meaning fellow he is in canon and not be TOO much "darker and edgier" as a result of a decade-long association with Euron.)
And in the meantime, the Others are stirring in the lands beyond the wall. Not sure how the dragons will be born in this scenario. The broad strokes of the canonical scenario could still take place, albeit in different circumstances. That would deprive the planned Targaryen restoration of its Dothraki manpower and could even cause infighting among its surviving leaders--Viserys will likely try to claim Danaerys as his wife, but Danaerys, now the Mother of Dragons, may not be interested. There's also "Aegon" and someone from House Martell (like Quentyn or if Doran learns about Danaerys' bad-boy fetish, maybe Oberyn) or even Euron who would also like that honor.
Anyone want to write a fan-fic set in this world? I'll blog about it if credit is given. Here's the second part of the timeline, bringing us to this world's version of the War of the Five Kings.