The Dragon has Three Heads
"Three heads of the dragon... yes... but the third will not necessarily BE a Targaryen..."
~ GRRM @ at some convention
The conclusion is simple. The heads of the dragon refer to people and while the two are Targaryens, the third might not be one.
We can ignore anything that says this group of things are of different things (for example, I've seen "two Targaryens plus a concept"), as that makes no narrative sense. The key is exactly that last sentence, the third one may not be a Targaryen, which implies the three heads are all people and that two are Targaryens and the third is not quite one. Therefore, these three dragon heads must refer to people associated with dragon, with two of them being Targaryens but the last one not necessarily being one.
The most popular theory is that the two Targaryens are Danerys Targaryen and Jon Snow, with the third being the youngster that claims to be Aegon VI Targaryen and appears first in ADWD. This identification ignores the narrative framework for both Aegon VI Targareyn and Jon Snow.
On one hand, Aegon is often associated with kingly imagery (for example, the chapter he's introduced starts with six chests, the kid himself is introduced as standing at a higher ground than the rest, and ends with a turtle who is said to witness the birth of kings), he's accompanied by a lot associated with his parents such as Jon Connington (his father's hand of the kind) and some dornish / royne people (his mother's land and culture). Another thing to note is that Varys introduces him as the real thing to a dying man.
Moreover, there is at least one "baby switch" story that shadows this one, a prince baby being switched with a nobody, sent away to protect against a Baratheon. In specific, Mance Rayder's son (the wildling "king") being switched with Gilly's son, then sent South for protection against another Baratheon, which is notably a plan concocted by Jon Snow (another of Rhaegar's kids, as if preparing him and the reader to "believe" such a scenario is possible).
On the other hand, Jon Snow's core character revolves around two facts, that he is a bastard and that he loves his Stark family. The reveal that he's not Eddard Stark's bastard but Lyanna Stark's child doesn't erase the latter, as he's a Stark through his mother. Still, the nature of Rhaegar and Lyanna's relationship may erase the former. If Jon is illegitimate, that doesn't erase the former, but if he's legitimate somehow (Targaryens are said to take multiple wives), that erases the former and replaces it another. It can go either way.
Moreover, there is at least one "legitimized bastard" story that shadows this one. In specific, Jon Snow is offered to be legitimised both by Stannis Baratheon (something that is a true temptation, as it would give him everything he secretely longed for all his life), but also by Robb's will as it names Jon Snow as his heir over sisters and that's only possible through legitimising him. It's my conviction that "the rule of three" applies, therefore that Jon will reject Robb's will like he rejected Stannis' offer, but will have a third legitimization opportunity and that this time around he'll acept.
To be more specific and in contrast with Aegon, who's introduced with kingly imagery, Jon is introduced with bastardy imagery. Bran introduces him into the narratve as his bastard brother, while Jon's first POV chapter starts with him musing that he's a bastard. While Jon has "kingly" imagery, it doesn't come associated with imagery from his father's side, and seems to be self-contained to the North (for example, the first inside joke is "kings hiding under the snow" or Mormont's crow calling him king while he's at the wall). Jon's hidden parentage comes along with prince imagery instead (for example, the anti-parallel with bastard prince Joffrey). With my conviction explained above, I do believe it will come to Aegon legitimising Jon as his heir until he has kids (a parallel to Robb's will). Most (if not all) foreshadowing falls into place. Aegon VI is king, Jon is the (bastard) prince. An example would be Sansa's "Glory to your betrothed," Ser Arys answered at once. (...) "He is the dragon's heir." which fits with Jon as Aegon's heir.
Combined, this interpreation suggests that the popular theory is actually backwards: Aegon VI is the real thing while Jon Snow is the Blackfyre (bastard Targaryen). This is in accordance to the way the text is presented.
"THe dragon has three heads" is referenced for the first time to Danerys Targaryen in a prophetic inducing Shade of the Evening tripping out at the House of Undying.
The man had her brother's hair, but he was taller, and his eyes were a dark indigo rather than lilac. "Aegon," he said to a woman nursing a newborn babe in a great wooden bed. "What better name for a king?"
"Will you make a song for him?" the woman asked.
"He has a song," the man replied. "He is the prince that was promised, and his is the song of ice and fire." He looked up when he said it and his eyes met Dany's, and it seemed as if he saw her standing there beyond the door. "There must be one more," he said, though whether he was speaking to her or the woman in the bed she could not say. "The dragon has three heads." He went to the window seat, picked up a harp, and ran his fingers lightly over its silvery strings. Sweet sadness filled the room as man and wife and babe faded like the morning mist, only the music lingering behind to speed her on her way.
I'll say that whatever the House of Undying shows, may not be reality. Rhaegar wanting to fulfill some prophecy with three kids is not referenced anywhere else, except this moment which is the equivalent of a very bad drug trip. It's worth mentioning though, Ratgar saw a comet in the sky and thought he should impregnate his wife, against medical advice because she who was recovering from giving birth his first child. It could be, but it could not be.
Regardless, what's important to note is what's being prophetized. Much like GRRM's convention remark, Rhaegar identifies the three heads as people. So far so good. However, he also gives us an order: omitted Rhaenys as she was born already, Aegon in mother's lap, Danerys when Rhaegar looks up to "see" her at the door, then finally Jon when he says "there must be one more". On one hand, Rhaenys was murdered and Danerys is in this as well, so the conclusion is that the former "replaced" the latter in the prophecy. On the other hand, if this had been a real memory, than Rhaegar would have two legitimate kids at the time (Rhaenys and Aegon) such saying "there must be one more" suggests a third child (Jon), which goes well with GRRM saying "the third may not be a Targaryen". So in order, we have Aegon, Danerys, Jon.
Danerys later reflects upon what this prophecy means and comes the conclusion that these heads are supposed to be people. This is because the Targaryen coat-of-arms is a dragon with three heads, each head representing three Targaryens.
"The dragon has three heads," she sighed. "Do you know what that means, Jorah?"
"Your Grace? The sigil of House Targaryen is a three-headed dragon, red on black."
"I know that. But there are no three-headed dragons."
"The three heads were Aegon and his sisters."
"Prince Aegon was Rhaegar's heir by Elia of Dorne," Ser Jorah said. "But if he was this prince that was promised, the promise was broken along with his skull when the Lannisters dashed his head against a wall."
"I remember," Dany said sadly. "They murdered Rhaegar's daughter as well, the little princess. Rhaenys, she was named, like Aegon's sister. There was no Visenya, but he said the dragon has three heads. What is the song of ice and fire?"
"Your Grace," he conceded, "the dragon has three heads, remember? You have wondered at that, ever since you heard it from the warlocks in the House of Dust. Well, here's your meaning: Balerion, Meraxes, and Vhagar, ridden by Aegon, Rhaenys, and Visenya. The three-headed dragon of House Targaryenâthree dragons, and three riders."
When Brown Ben left, she lay back on her cushions. "If you were grown," she told Drogon, scratching him between the horns, "I'd fly you over the walls and melt that harpy down to slag." But it would be years before her dragons were large enough to ride. And when they are, who shall ride them? The dragon has three heads, but I have only one.
"No dragon has ever had three heads except on shields and banners," Armen the Acolyte said firmly. "That was a heraldic charge, no more. Furthermore, the Targaryens are all dead."
"Not all," said Alleras. "The Beggar King had a sister."
"The dragon must have three heads," he wailed, "but I am too old and frail to be one of them. I should be with her, showing her the way, but my body has betrayed me."
The prophecies in ASOIAF are always misunderstood. Danerys is no exception, as she's wrongly identifing people and their motives. One thing is for sure in all these mentions though, the "dragon heads" are meant to be people and one of them is Danerys.
Danerys thinks these three dragons are supposed to mimic the original trio, with herself as Aegon and two men she'll take as lovers as the two sister wives. This is where the misunderstanding is, because it's obvious from the framework that is backwards.
These at least she could rely on, or so she hoped . . . and Brown Ben Plumm as well, solid Ben with his grey-white hair and weathered face, so beloved of her dragons. And Daario beside him, glittering in gold. Daario and Ben Plumm, Grey Worm, Irri, Jhiqui, Missandei . . . as she looked at them Dany found herself wondering which of them would betray her next.
The dragon has three heads. There are two men in the world who I can trust, if I can find them. I will not be alone then. We will be three against the world, like Aegon and his sisters.
Danerys is a dumb bitch and the text shows us exactly how. The idea of Danerys being betrayed comes together with the other two heads. They're not lovers, they're betrayers. It occurs again in ADWD as she's in the Dothraki Sea, contemplentanting if the "king" betrayed her and a wolf answers in the distance.
The framework fits this foiled scenario. The original trio was a man married to both his sisters, but only he became king (later, only the man could rule according to Targaryen law). In contrast, Danerys is a woman (foil) and both her brothers are dead (foil), whom are replaced with nephews that have a bigger claim than her (foil) and whom will not be involved / married with her (foil).
It's also thematically relevant and poignant for the trios to be foiled. The Targaryen king dynasty started with an alliance between three dragons, it's fitting that it ends with a war between three dragons. It's what they've been threatening all along with the Dance of the Dragons after all.
