Spoiler Alert: This piece may contain spoilers from any material of A Song of Ice and Fire.
Many have wondered how the beloved Rhaegar Targaryen could be so different from his mad and cruel father, Aerys II Targaryen. It is a fair question. After all, Aerys's other son, Viserys, clearly did not fall so far from the tree. Despite Robert's hateful claims, the rest of the seven kingdoms had the highest praise for Prince Rhaegar. How can such a disconnect be reconciled?
The answer is actually quite simple: Rhaegar is not the son of Aerys II. The evidence for this is so, umm, stark that I am surprised I was unable to find any direct discussion of the matter. Let us begin by taking a look at the timeline of births from Aerys II and Rhaella:
263 AC, a miscarriage.
264 AC, a miscarriage.
267 AC, stillbirth of Shaena Targaryen.
269 AC, birth of Daeron Targaryen. Died half a year later.
270 AC, stillbirth of an unnamed child.
271 AC, a miscarriage.
272 AC, birth of Aegon Targaryen, two months premature. Died in 273 AC.
274 AC, birth of Jaehaerys Targaryen. Died later that year.
275 AC, King Aerys II blames his infidelity for these misfortunes and finally vows to be faithful to his wife.
276 AC, birth of Viserys Targaryen, small, robust, and quite healthy.
284 AC, birth of Daenerys Targaryen in exile on Dragonstone.
Which of the above does not fit the pattern? While one could certainly raise legitimate questions about Daenerys, Rhaegar's birth is undeniably out of place as the only healthy child before Aerys's vow of fidelity. It's worth noting that Aerys II was born in 244 AC and Rhaella in 245 or 246. They were quite young, albeit not impossibly young, to give birth to Rhaegar.
Why did Aerys attempt to have children so many times after so many failures when Rhaegar was everything a king could desire in an heir? If the answer is that he wanted many children, then why the long wait after Viserys?
Furthermore, after Joanna Lannister gave birth to Jaime and Cersei, Aerys said, "I appear to have married the wrong woman." This remark makes little sense if, by that time, each woman had given birth to one trueborn son.
It also seems that Aerys deliberately married Rhaegar to someone who seemed unlikely to bear children: "Jon Connington remembered Prince Rhaegar’s wedding all too well. Elia was never worthy of him. She was frail and sickly from the first, and childbirth only left her weaker. After the birth of Princess Rhaenys, her mother had been bedridden for half a year, and Prince Aegon’s birth had almost been the death of her. She would bear no more children, the maesters told Prince Rhaegar afterward."
I submit that Viserys is the first legitimate child of Aerys, so let us zero in on his birth for a moment. Even though Viserys was healthy, Aerys II was extremely, insanely paranoid for his safety. Nobody, not even Rhaella, was allowed to touch him. He could not even be alone with his mother. Aerys commanded that his food taster suckle the breasts of the wet nurse in case of poison. The child's gifts were burned for fear of sorcery. When Tywin held a tournament to celebrate the prince's birth, Viserys and Rhaella were commanded to remain in King's Landing. This is the behavior of a mad king who has long been deprived of a direct heir.
It is important to note that, in contrast, Rhaegar was permitted to compete in said tournament. Aerys was not stupid enough to openly disown Rhaegar now that he had a legitimate son, but he evidently had no qualms about the young man engaging in such risky activities.
A year after the birth of Viserys, Aerys believed Rhaegar conspired with Tywin to have him killed at the Defiance of Duskendale. The fact that the king became suspicious of the prince immediately after the birth of Viserys is quite telling. I'm reminded of how Prince Caspian, nephew of a king, had to flee once a son was born.
This theory answers many questions, but asks an obvious one: who, then, were Rhaegar's parents? It seems fitting to me that such a beloved character as Rhaegar would be more closely related to Aegon V, our favorite Targaryen from the Dunk and Egg stories, than originally claimed. I believe the best explanation is that Rhaegar was not Egg's great-grandson, but his grandson through Duncan the Small, Prince of Dragonflies. We are not told whether Duncan and Jenny had any children, as the record keepers seem to have lost interest in them after the prince gave up the throne and married for love.
What we do know is that Jenny had a friendship with a prophetess, the woods witch who is probably the Ghost of High Heart who "gorged on grief at Summerhall." We also know that Rhaegar had a passion for prophecy. Egg and Duncan the Small died at Summerhall trying to fulfill a prophecy about the rebirth of dragons. Rhaegar often returned to Summerhall in solitude to compose songs of sadness. Rhaegar felt a stronger connection to those lost in the fire than the other survivors who actually knew them in life.
"And yet Summerhall was the place the prince loved best. He would go there from time to time, with only his harp for company. Even the knights of the Kingsguard did not attend him there. He liked to sleep in the ruined hall, beneath the moon and stars, and whenever he came back he would bring a song. When you heard him play his high harp with the silver strings and sing of twilights and tears and the death of kings, you could not but feel that he was singing of himself and those he loved."
I suspect that both Jenny and Rhaella were pregnant at the same time. When tradegy struck Summerhall, I believe the baby almost "...died, but for the valor of the Lord Comman..."der. (The History Of Archmaester Gyldayn) Ser Duncan the Tall and Jenny of Oldstones either died or left in grief. Rhaella, being consistent with her early pattern, either miscarried or had a stillbirth. Aerys and Rhaella then took Jenny's child as their own.
Why adopt Rhaegar? There are many possible reasons. Perhaps Jenny abandoned her child to their care. After all, what we know of Jenny's Song suggests that she went mad if she lived. Perhaps Aerys or Rhaella took pity on their infant cousin, who was family after all. Perhaps Aerys thought it unfair that his own child was not the fulfillment of the prophecy and took the resulting baby for himself. Perhaps Rhaella was heartbroken from her first stillbirth, or even lied to her husband at the start. The imagination need not stretch at all.
This conclusion also ties closely with events later in Rhaegar's life. It shows a connection between the prince who crowned Lyanna Stark Queen of Love and Beauty and his father and grandfather who married for love at great cost to the realm. It gives us another reason why Rhaegar was probably plotting to overthrow Aerys II around the time of the Tourney at Harrenhal. Perhaps Rhaegar's historical research even gave him reason to suspect Aerys of foul play at Summerhall.
Finally, this theory is more poetic for the readers who have a love of Dunk and Egg and who also believe that Rhaegar still has a heroic part to play in the Song of Ice and Fire.
"My father was only nine-and-thirty. He had it in him to be a great king, the greatest since Aegon the Dragon. Why would the gods take him, and leave you? Begone with you, Ser Duncan. Begone."
Proponents of the theory that the dragon flew in disguise to the wall would do well to recall Dunk's thoughts at the opening of the tale: "Why would they name it a dragonfly? he wondered. It looks nothing like a dragon."
Additions and corrections welcome in the comments!