Rhaegar wins (and consequences)
This is something that I've been wanting to write about for some time and would like to hear your thoughts about it.
The overly simplified version is that Rhaegar wins at the Trident by defending the ford instead of attempting a cross. Robert is injured by the time he meets, and is promptly slain, by Rhaegar. Lord Frey arrives late and entraps the rebel army, capturing Ned, Jon Arryn and Hoster. Rhaegar immediately marches back to King's Landing with an army on his back and meets Tywin there. Aerys still attempts the wildfire plot and is slain by Jamie Lannister as in canon.
From that point onwards, I am mostly doing educated guesses and assumptions that I hope are in character:
Stannis eventually surrenders as Robert was dead and the war lost.
Lyanna still dies giving birth to Jon (or rather whichever Targaryen names that fits him Aemon, Valarr, Jaehaerys) as from what I've seen some people talking, Lyanna was most likely going to die even if cared by a better prepared Maester.
If Rhaegar is smart enough he takes Ned alongside some Northern lords to see Lyanna and spins a tale of how his father wished to kill her due to the Knight of the Laughing Tree at Harrenhal.
With Lyanna dead and his three heads of the dragon in hand he had no reason to set Elia aside, instead I see him legitimizing Jon to placate the Starks and then making amends to Dorne.
When it comes to punishing the rebels, I don't think he will do anything as extreme as executions or even the wall. His father had punished the rebels in Duskendale in a extreme manner and, if Rhaegar cares about his reputation, I'd see him trying to send a clear message that he's not Aerys.
I believe he may punish the North simply by taking Benjen and some minor heirs as hostages to ensure future compliance.
Likewise for the Vale, with him taking Harry as hostage, perhaps with the added effect of decreeing that the loyalist Graftons are either spared of paying taxes to the Eyrie or compensated for their losses in the talking of Gulltown, maybe even both.
The Riverlands are a trickier case as Hoster didn't lose an immediate family member as Ned and Jon Arryn did. I remember a Mallister was slain alongside Brandon but I don't think that's a good justification for declaring for Robert. Moreso, unlike the Arryns and Starks, the Tullys never managed to consolidate their hold into the Riverlands that well, and are easier to punish without incurring the wrath of their bannermen. I don't think Rhaegar revokes Riverrun but I can see him expanding the Crownlands by taking direct allegiance from houses that were loyal to the Targaryens (Darrys, Mootons, Rygers, Goodbrooks and most likely the Whents). He could also revoke the paramountcy and reward the Freys, or at least offer them tax breaks.
I can either see him sending Stannis to the wall or simply taking Renly hostage as he did with the North. Since there was already the Laughing Storm Rebellion a few decades ago, I can also see him giving them the Riverlands treatment and incorporating the loyalist lords into the Crownland (Fell, Cafferen, Grandison).
Tywin would most likely require that Jaime is released from his vows, and I don't think that Rhaegar would have him executed or sent to the wall should the truth of the wildfire plot comes to light.
Rhaegar's reign wouldn't be stable, as lords would still remember that the whole mess started with him kidnapping Lyanna Stark. He would be a better ruler than Robert, but with less goodwill due to his reputation, and perhaps lesser skilled councillors. (Jon Connington instead of Jon Arryn, Paxter Redwyne instead of Stannis, a Grafton or Mooton as Lord of Laws, Varys probably escapes leaving space for Oberyn).
Conflict would come in the form of the Lannisters and Tyrells competing to put their blood in the throne, a lesser cooperative North facing Mance Rayder, Blackfyre shenanigans with Ilyrio and Varys and the good old conspiring lords seeking to install either Jon or Viserys in the throne for their own gains.
Am I missing something?