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Dragon dreams affect those with the blood of the dragon, who are known to have premonition-like dream ability - they are no ordinary dreams. They often involve dragons but not always.
Dragon dreams appear to have haunted those with the blood of the dragon throughout the generations; some more than others.
Daenys The Dreamer
When Daenys Targaryen was still a maiden, she had visions, including a powerful prophetic dream that Valyria would be destroyed. Her father had enough faith in her prophetic abilities to relocate his family to Dragonstone. Valyria and all the other dragonlords were destroyed twelve years later.
Viserys was hitting her, hurting her. She was naked, clumsy with fear. She ran from him, but her body seemed thick and ungainly. He struck her again. She stumbled and fell. He struck her again. She stumbled and fell. “You woke the dragon,” he screamed as he kicked her. “You woke the dragon, you woke the dragon.” Her thighs were slick with blood. She closed her eyes and whimpered. As if in answer, there was a hideous ripping sound and the crackling of some great fire. When she looked again Viserys was gone, great columns of flame rose all around, and in the midst of them was the dragon. It turned its great head slowly. When its molten eyes found hers, she woke, shaking and covered with a fine sweat.
There was only her and the dragon. Its scales were black as night, wet and slick with blood. Her blood, Dany sensed. Its eyes were pools of molten magma, and when it opened its mouth, the flame came roaring out in a hot jet. She could hear it singing to her. She opened her arms to the fire, embraced it, let it swallow her whole, let it cleanse her cleanse her and temper her and scour her clean. She could feel her flesh sear and blacken and slough away, could feel her blood boil and turn to steam, and yet there was no pain. She felt strong and new and fierce.
After the birth of her three dragons, Daenerys Targaryen seems to no longer be dreaming of the black dragon, but she is plagued by other dreams and visions.
That night she dreamt she was Rhaegar, riding to the Trident. But she was mounted on a dragon, not a horse. When she saw the Usurper’s rebel host across the river they were armored all in ice, but she bathed them in dragonfire and they melted away like dew and turned the Trident into a torrent. Some small part of her knew that she was dreaming, but another part exulted. This is how it was meant to be. The other was a nightmare, and I have only now awakened.
As Queen of Meereen she chains Viserion and Rhaegal in the darkness of the Great Pyramid's pit and Drogon is lost to her in the Dothraki Sea. She is finding herself vexed by nightmares, and her dreams have changed, no longer offering her succour:
Beneath her coverlets she tossed and turned, dreaming that Hizdahr was kissing her … but his lips were blue and bruised, and when he thrust himself inside her, his manhood was cold as ice.
As he weakens he spends more time sleeping. Sometimes he would call for Sam, insisting that he had to tell him something, but he would often forget what he meant to say by the time Sam arrived. Even when he did recall his talk was all a jumble. He spoke of dreams and never named the dreamer, of a glass candle that could not be lit and eggs that would not hatch. He said the sphinx was the riddle, not the riddler. He asked Sam to read for him from a book by Septon Barth, whose writings had been burned during the reign of Baelor the Blessed. Once he woke up weeping, wailing:
He mentions "the prophecy" and his "brother’s dream" to Sam - but does not mention what prophecy or which brother to Sam, or what the dream entailed. 
Prince Daeron Targaryen told Dunk about a dream he had. A great red dragon would fall on Ser Duncan, dead, but the knight would walk off alive. He said he did not know what it meant, but that his dreams came true and that he was scared of Dunk. The dream turned out to be metaphorical, referring to the death of Prince Baelor Targaryen.
Prince Aegon Targaryen mentions another dream of Daeron's:
Daemon II Blackfyre
Daemon II Blackfyre, in the guise of 'John the Fiddler', talks to Duncan, saying that he recognized him on the road. Duncan had appeared to him in a dream in which Duncan wore the all-white armor of the Kingsguard. The Fiddler says his dreams always come true, as he dreamt his brothers dead once and also a dragon hatching from an egg at Whitewalls.
- A Feast for Crows, Chapter 11, The Kraken's Daughter.
- The World of Ice & Fire, The Reign of the Dragons: The Conquest.
- A Game of Thrones, Chapter 11, Daenerys II.
- A Game of Thrones, Chapter 23, Daenerys III.
- A Storm of Swords, Chapter 23, Daenerys II.
- A Dance with Dragons, Chapter 43, Daenerys VII.
- A Feast for Crows, Chapter 26, Samwell III.
- A Feast for Crows, Chapter 35, Samwell IV.
- The Hedge Knight.
- The Mystery Knight.