When his father Eddard Stark is executed, Bran will see the shape of doom descending on all of them, but nothing he can say will stop his brother Robb from calling the banners in rebellion. All the north will be inflamed by war. Robb will win several splendid victories, and maim Joffrey Baratheon on the battlefield, but in the end he will not be able to stand against Jaime and Tyrion Lannister and their allies. Robb Stark will die in battle, and Tyrion Lannister will besiege and burn Winterfell. -GRRM, 1993

When GRRM's 1993 letter outlining (or as he preferred, sharing "broad notions") the first book of ASOIAF leaked a decade ago the Robb vs. Joffrey duel was often singled out as an especially ludicrous departure from the published novel. But in the years since diligent fans have discovered that the chapters of AGOT written at that point were actually very similar to their final versions.

Live Steel

The encounter in the training yard between Robb and Joffrey seems to strongly foreshadow the future duel mentioned in the outline:

“Robb may be a child,” Joffrey said. “I am a prince. And I grow tired of swatting at Starks with a play sword.”

“You got more swats than you gave, Joff,” Robb said. “Are you afraid?” Prince Joffrey looked at him. “Oh, terrified,” he said.

Ser Rodrik tugged thoughtfully at his white whiskers. “What are you suggesting?” he asked the prince.

“Live steel.”

“Done,” Robb shot back.

“You’ll be sorry!” -Arya I, AGOT

Important to note that originally Robb and Joffrey were the same age (12) here, only much later in writing AGOT did GRRM age up Robb to 14.

Arya could see Robb bristle. His pride was wounded. He turned on Ser Rodrik. “Let me do it. I can beat him.”

“Beat him with a tourney blade, then,” Ser Rodrik said.

Joffrey shrugged. “Come and see me when you’re older, Stark. If you’re not too old.”

There was laughter from the Lannister men. Robb’s curses rang through the yard. Arya covered her mouth in shock. Theon Greyjoy seized Robb’s arm to keep him away from the prince. Ser Rodrik tugged at his whiskers in dismay.

Joffrey feigned a yawn and turned to his younger brother. “Come, Tommen,” he said. “The hour of play is done. Leave the children to their frolics.”

That brought more laughter from the Lannisters, more curses from Robb. Ser Rodrik’s face was beet-red with fury under the white of his whiskers. Theon kept Robb locked in an iron grip until the princes and their party were safely away. -Arya I, AGOT

This scene establishes a very personal and bitter rivalry between Robb Stark and Joffrey Lannister Baratheon. But in the published books, this tension is unresolved and neither character ever meet again.

Kill your brother myself

Joffrey fighting on the battlefield might seem a dubious prospect, but AGOT ends with Joffrey stating his intention to raise an army and march out to meet Robb Stark in battle. First, he harks back to the abortive duel with Robb Stark and row over live steel at Winterfell:

When he smiled, she knew he was mocking her. “Your brother is a traitor too, you know.” He turned Septa Mordane’s head back around. “I remember your brother from Winterfell. My dog called him the lord of the wooden sword. Didn’t you, dog?” -Sansa VI, AGOT

Joffrey then promises to raise a new Lannister army and kill Robb Stark himself:

Joffrey gave a petulant shrug. “Your brother defeated my uncle Jaime. My mother says it was treachery and deceit. She wept when she heard. Women are all weak, even her, though she pretends she isn’t. She says we need to stay in King’s Landing in case my other uncles attack, but I don’t care. After my name day feast, I’m going to raise a host and kill your brother myself. That’s what I’ll give you, Lady Sansa. Your brother’s head.”

A kind of madness took over her then, and she heard herself say, “Maybe my brother will give me your head.” -Sansa VI, AGOT

Shortly afterwards at his name day celebrations Joffrey shares how he intends to challenge Robb Stark to single combat:

He laughed. "That's funny, don't you think? The dragon was their sigil. It's almost as good as if some wolf killed your traitor brother. Maybe I'll feed him to wolves after I've caught him. Did I tell you, I intend to challenge him to single combat?"-Sansa I, Acok

Note this chapter was likely originally written for AGOT, as several of the early chapters of ACOK were cut from the previous book.

Of course, after his name day Joffrey doesn't end up scraping together an army and trying to kill Robb Stark.

TL,DR; GRRM was originally planning to have Robb Stark and Joffrey "Baratheon" meet on the battlefield and have it end badly for Joffrey. There are still possible remnants of this plot in the published text.