The Battle Above the Gods Eye
Aemond Targaryen on Vhagar ÔÇö the oldest, largest dragon left in the world ÔÇö against Daemon Targaryen on Caraxes, over a lake at the heart of the realm. Both dragons and both riders are lost together. It is the single most costly exchange of the war and settles nothing at all.
The fuller tale
Aemond Targaryen had been burning the riverlands for months with Vhagar, the oldest and largest dragon in the world, and nothing the queen's party could put in the air was a match for her. Daemon Targaryen took Caraxes to Harrenhal, sat in the great empty castle Aemond had been using as a base, and waited for him to come back.
The story that has come down is that the two of them agreed to it ÔÇö that Daemon sent word and Aemond answered, and both understood what it meant. They met over the Gods Eye at dusk, above the lake where the Pact had been sworn eight thousand years earlier, and the whole of both armies stood on the shore and watched.
Vhagar was three times Caraxes's weight. Daemon's answer to that was to give up any thought of surviving: he brought Caraxes onto her from above, out of the low sun, jumped the gap between saddles in the air and drove his sword through Aemond's remaining eye. All four came down into the lake together. Aemond's body was never found; Daemon's was never found either, and there are people in the riverlands who still say he walked out.
The most expensive exchange of the war
Two riders and two dragons in one evening, including the last of the great beasts from the Conquest. Vhagar had carried Visenya Targaryen over the Field of Fire. She was a hundred and eighty years old and there would never be another.
It settled nothing. Aemond's death removed the greens' one unanswerable weapon and Daemon's removed the blacks' one competent commander, and the war ground on for another year with less to fight it with. If the Dance has a single image, it is this one: the two best pieces on the board taken off it at once, over the lake where men and children once swore to share a country.
Elsewhere in the wiki
 See the world as it stood in 130 AC What became of it  if an ending is written, it is here. Spoilers.
The chronicle records no ending here ÔÇö or none the maesters have set down.