Some notes since people repeatedly bring it up. On incest – the world of Westeros follows Medieval European standards. Therefore, taboo relationships include siblings, parent-child, and aunt-nephew/uncle-niece. First cousins, however, are NOT considered incestuous (good news for all those Jon/Sansa 'shippers). While Westeros history is obviously rife with incestuous relationships anyway, especially the Targaryens, that doesn't mean that everyone is cool with it happening. And we can assume that the modern Starks, with their strict moral code, would be uncomfortable with the concept, regardless of what happened in their family tree several generations earlier. Basically, you can assume that the immediate fall-out from the reveal of Jon's parentage isn't going to be particularly pleasant or easy. On dragons, I can find no record of any non-Valyrian ever riding a dragon, which in Westeros means Targaryen. So Jon's newfound dragon-riding abilities are bound to make people suspicious even if they don't immediately guess who his parents are. This chapter may not make everyone happy, but fear not - the story isn't over yet.
After some consideration, Jon brought Rhaegal down to land in the same hunting clearing they'd taken off from, days before. He'd originally intended to fly straight back to Winterfell after dealing with the invasion of Last Hearth, but had realized that he needed to take the opportunity to survey the North and supervise the evacuation. The situation was bleak. Wights were roaming everywhere, those who'd escaped the invasion risked starvation and death from the cold, and supplies were rapidly running out. Perplexingly, Jon hadn't seen or heard of Viserion since he'd taken down the Wall at Eastwatch and the absence worried him. The Last River appeared to be holding back the tide of the dead for now, but it grew colder every day. Once it froze over or the dead managed to break through the mountain passes to the west, the rest of the North would be in peril.
Jon regretfully slid down Rhaegal's back. For all that his mission had been grim, he'd enjoyed having a clear purpose, uncomplicated by family and conflicting emotions. And he'd enjoyed flying and Rhaegal's company. He understood now, how Daenerys thought her dragons beautiful. There was a depth of feeling, an intelligence that even surpassed Ghost. With each passing day in Rhaegal's company, warging into him had begun to feel less foreign, less awkward.
"I'll miss you," Jon told the dragon while petting his nose. Rhaegal snuffled at him and Jon could feel the dragon's confusion. Jon leaned his forward against the scales, wishing he could avoid what was to come.
Rhaegal swung his head up rapidly, alerting Jon to the presence of another in the clearing. Daenerys strode towards him and Jon knew that his vain hope of secrecy had been dashed. She looked angry and glorious and her presence awoke all the feelings he'd been trying to convince himself that he didn't have for the last few days.
"Daenerys. I'm glad to see you're back safely," Jon began.
"And I you. Despite stealing my dragon to go hunt wights." Daenerys sounded as if she was trying for light-hearted, but landed somewhere on the side of bitter.
Jon shifted uncomfortably, feeling as if he was once again a child and in trouble for stealing the hotcakes. "Daenerys, I needed to get North quickly. This was the only way."
Daenerys closed her eyes and swallowed. "I know. I do know. But Jon, you don't know what you've done."
"We're back safely. I kept Rhaegal from harm. He's still yours," Jon hesistantly reassured Daenerys.
Daenerys shook her head sadly. "Did you think dragons were like horses? Borrow one from the stable and return it? They're not. Dragons accept one rider for both of their lifetimes. Once a dragon has bonded, the only way to break that bond is death."
Jon blanched. "Daenerys, I didn't know that. I never meant …"
"Targaryens have not flown dragons in centuries, but we remember. Viserys passed on the lore to me as he heard it from our mother. You did not know what you should have, but what is done is done. Rhaegal belongs to you now, and all that goes with it." Daenerys stood only a few feet away now, but Jon felt a chasm between them. "Shall I call you Aegon then?"
Jon recoiled from the acknowledgment of his true name. "No! No," he repeated. "That name is but a story. The life that went with it died with my …" Jon paused. Rhaegar may have sired him, but his father would always be Ned Stark. "It died. Let it stay dead."
"But you're not dead. You've chosen a dragon, a better proof of who you are than any vision or piece of paper ever will be. To survive this war, Westeros needs both of its dragons. Together, we can rule. I need you by my side, as my king." Daenerys stepped closer to him and despite the cold bite of the air, Jon felt as warm as if he were inside an inferno. Part of him, the primitive shameful part, wanted to grab her and drive into her until they'd both reached oblivion.
"I can't," Jon said, his voice cracking. "You know we can't."
"We can. We must," Daenerys insisted.
Jon looked away from her. "Daenerys, it's forbidden."
"My family, our family, has never abided by the rules of musty old religious texts." She placed a hand on his arm. "Jon, please look at me. Knowing who you are doesn't change how I feel."
Jon looked back at her reluctantly. "It doesn't matter how we feel. What we want … the gods banned it for a reason. If you bore my child …" Daenerys made a noise as if to interrupt, but Jon spoke over her. "If you bore my child, who's to say whether he would carry the taint? Targaryens have been punished with insanity and cruelty before for their transgressions."
Daenerys withdrew as if struck. "Is that how you view me? Tainted?"
"No! That was … a poor choice of words. But breaking the law of the gods has consequences, Daenerys. I could not bear a child of mine to be cursed like the Mad King." Jon pleaded with Daenerys with his eyes. He needed her to understand why he couldn't be with her.
Daenerys's face might have been stone except for two spots of color that appeared high on her cheeks. "What then? Will you seize the Iron throne? Ask me to bend the knee?"
"Of course not. I couldn't ask that of you." Jon felt his control slipping away from him as if he was teetering on the edge of a very dark abyss.
"You should. If we do not rule together then we do not rule at all. Did you imagine that your heritage would stay hidden while you flew over half the North on my dragon?" Daenerys's eyes blazed.
"No one knows! No one needs to know!" Jon shouted back.
"You did not consider how your proud Northern Lords would react if they knew their own chosen king was their rightful king! They do not support my claim even now. Even after you already renounced your rule and they still think you Ned Stark's bastard!" Daenerys shouted back.
"I keep my word," Jon ground out.
"It doesn't matter! We have rival claims to the throne and while the South may yet support me, the North will surely support you. Will you return to the Night's Watch? Because nothing less will bring the North into my domain," Daenerys demanded.
"I can't do that. You can't ask me to do that." Jon recoiled from the thought. His heart pounded painfully.
What Daenerys meant to say next, Jon couldn't say, because Rhaegal brought down one huge wing between them and actually hissed at her. Overhead, Drogon let out a warning cry. Jon placed his hand on Rhaegal's side and immediately felt a wave of confusion and distress emanating from the dragon. Jon forced himself to breath evenly until he could project something resembling calmness. Rhaegal slowly relaxed and lowered his wing revealing a white-faced Daenerys staring at them.
"I see. I have already lost." Daenerys crossed her arms protectively in front of herself.
Jon wanted to say something to comfort her. He wanted to hold her and forget that the last few days had ever occurred. But instead, he just watched as she walked away. He needed to let her go.