Why are there no other dragons in ASOIAF?

Question

In order to hatch a dragon in A Song of Ice and Fire, the eggs must be heated to an incredible temperature, as shown when Daenerys' dragons are hatched. So why aren't there any other dragons?

Eggs can be found (albeit rarely) in the Shadow Lands by Asshai and one can assume that other people have eggs. So why haven't any of them, even accidentally, been hatched by their owners?

Is there an in-universe reason?

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_{I've read all the main books, watched up and including Season 6 and read The World of Ice and Fire so "spoilers" are fine.}

Answer 1

Flawed Premise

In order to hatch a dragon in A Song of Ice and Fire, the eggs must be heated to an incredible heat

Who said that? Dragon incubators do not exist in Planetos.

Hatching a dragon is not that simple.

Blood magic

All the accomplishments of the Dragon Lords were thanks to Blood magic. They possibly used blood magic to hatch dragons from Wyverns and certainly used blood magic to bind Dragons to themselves.

In Septon Barth's Dragons, Wyrms, and Wyverns, he speculated that the bloodmages of Valyria used wyvern stock to create dragons. Though the bloodmages were alleged to have experimented mightily with their unnatural arts, this claim is considered far-fetched by most maesters, among them Maester Vanyon's Against the Unnatural contains certain proofs of dragons having existed in Westeros even in the earliest of days, before Valyria rose to be a power.
_{TWOIAF - Beyond the Free Cities: Sothoryos}

also:

"What feeds a dragon's fire?" Marwyn seated himself upon a stool. "All Valyrian sorcery was rooted in blood or fire. The sorcerers of the Freehold could see across mountains, seas, and deserts with one of these glass candles. They could enter a man's dreams and give him visions, and speak to one another half a world apart, seated before their candles. Do you think that might be useful, Slayer?"
_{AFFC - Samwell V}

How did Daenerys do it?

The kind of Blood magic shown there is simple, you provide sacrifice of something and you get something in return.

"You'll have gold, horses, whatever you like."

"It is not a matter of gold or horses. This is bloodmagic, lady. Only death may pay for life."
_{AGOT - Daenerys VIII}

The eggs that Daenerys had were very old. They had turned into stones by the time they came into her possession. Fate of other dragon eggs would be similar since last living dragon died in reign of King Aegon III centuries ago.

Even Illyrio, who gave her the eggs, had no idea that they would hatch.

"I know that he gave me my dragon eggs."

He snorted. "If he'd known they were like to hatch, he'd would have sat on them himself."

That made her smile despite herself. "Oh, I have no doubt of that, ser. I know Illyrio better than you think. I was a child when I left his manse in Pentos to wed my sun-and-stars, but I was neither deaf nor blind. And I am no child now."
_{ACOK - Daenerys III}

Daenerys however figured out a way. She figured out the nature of Mirri Maz Duur's magic and cleverly improvised.

She provided sacrifice of:

1. Khal Drogo
2. A horse.¹
3. Mirri Maz Duur herself

"You will not hear me scream," Mirri responded as the oil dripped from her hair and soaked her clothing.

"I will," Dany said, "but it is not your screams I want, only your life. I remember what you told me. Only death can pay for life." Mirri Maz Duur opened her mouth, but made no reply. As she stepped away, Dany saw that the contempt was gone from the maegi's flat black eyes; in its place was something that might have been fear. Then there was nothing to be done but watch the sun and look for the first star.
_{AGOT - Daenerys X}

For coup de grâce, she climbed into the pyre herself. As a result, the stone eggs came back to life and dragons were hatched.

This was a one-time magical event. Only time it ever succeeded. GRRM cleared it up.

George_RR_Martin: Granny, thanks for asking that. It gives me a chance to clear up a common misconception. TARGARYENS ARE NOT IMMUNE TO FIRE! The birth of Dany's dragons was unique, magical, wonderous, a miracle. She is called The Unburnt because she walked into the flames and lived. But her brother sure as hell wasn't immune to that molten gold.

Not to mention, Daenerys might actually be someone very special. As Maester Aemon believed:

What fools we were, who thought ourselves so wise! The error crept in from the translation. Dragons are neither male nor female, Barth saw the truth of that, but now one and now the other, as changeable as flame. The language misled us all for a thousand years. Daenerys is the one, born amidst salt and smoke. The dragons prove it.
_{AFFC - Samwell IV}

Historical attempts

That does not mean there were no attempts to rehatch dragons from stone-eggs before. King Aegon III asked mages and other scholars from Essos to hatch dragons from his cache of eggs. It failed. Prince Aerion drank wildfire thinking that it would transform him into a dragon. It failed. King Aegon V tried to use wildfire to hatch dragons, it failed and House Targaryen was wiped out save one lonely branch of Jaehaerys II Targaryen.

The look Stannis gave her was dark. "Nine mages crossed the sea to hatch Aegon the Third's cache of eggs. Baelor the Blessed prayed over his for half a year. Aegon the Fourth built dragons of wood and iron. Aerion Brightflame drank wildfire to transform himself. The mages failed, King Baelor's prayers went unanswered, the wooden dragons burned, and Prince Aerion died screaming."
_{ASOS - Davos V}

And also:

There were dragons here two hundred years ago, Sam found himself thinking, as he watched the cage making a slow descent. They would just have flown to the top of the Wall. Queen Alysanne had visited Castle Black on her dragon, and Jaehaerys, her king, had come after her on his own. Could Silverwing have left an egg behind? Or had Stannis found one egg on Dragonstone? Even if he has an egg, how can he hope to quicken it? Baelor the Blessed had prayed over his eggs, and other Targaryens had sought to hatch theirs with sorcery. All they got for it was farce and tragedy.
_{AFFC - Samwell I}

And also:

What became of the dream of dragons was a grievous tragedy born in a moment of joy. In the fateful year 259 AC, the king summoned many of those closest to him to Summerhall, his favorite castle, there to celebrate the impending birth of his first great-grandchild, a boy later named Rhaegar, to his grandson Aerys and granddaughter Rhaella, the children of Prince Jaehaerys.

It is unfortunate that the tragedy that transpired at Summerhall left very few witnesses alive, and those who survived would not speak of it. A tantalizing page of Gyldayn's history—surely one of the very last written before his own death—hints at much, but the ink that was spilled over it in some mishap blotted out too much.

...the blood of the dragon gathered in one... ...seven eggs, to honor the seven gods, though the king's own septon had warned... ...pyromancers... ...wild fire... ...flames grew out of control...towering...burned so hot that... ...died, but for the valor of the Lord Comman...
_{TWOIAF - The Targaryen Kings: Aegon V}

Conclusion

There is no indication whether other people in possession of Dragon eggs ever attempted to hatch them. They would have thought it futile given that eggs were dead stones. They were used as valuable commodity, not as livestock. So in conclusion:

1. Blood magic is required to hatch dragons.
2. Existing eggs in current timeline are essentially stones.
3. Daenerys' feat was a one-time magical event, a miracle.

Also see: How many dragons have ever existed in Westeros and what became of them?

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^{1. It is possible that it may have been Rhaego's soul in Drogo (So Drogo would count as two) or Daenerys herself (Although she never died). Third sacrifice is unconfirmed as of now.}

Answer 2

Basically there is a running theme through the series that magic is on the rise. It's left to the reader to decide if the magic is returning because dragons have hatched or dragons have hatched because magic is returning.

I don't recall it being spelled out in the books as to why Dany was able to wake the eggs when other folks could not, but it does seem to have been hinted that it had to do with, not just with the fire, but also the blood magic associated with the live sacrifice of Mirri Maz Duur.

Answer 3

The top answer provided a great explanation of why there aren't eggs hatching around Planetos, but doesn't explain why the dragons went extinct in the first place. Two events are the main cause of the lack of dragons in-universe:

1. The Doom of Valyria:

The one thing that can be said for certain is that it was a cataclysm such as the world had never seen. The ancient mighty Freehold - home to drangons and to sorcerers of unrivaled skill - was shattered and destroyed within hours.

(...) even the dragons in the sky were engulfed and consumed.

(...) It is said that some Valyrian dragonlords in Tyrosh and Lys were spared, but that, in the immediate political upheavel following the Doom, they and their dragons were killed by citizens of those free cities. (The World of Ice and Fire, The Doom of Valyria)

So, when the Doom destroyed Valyria, most dragons went with it. The only survivors seem to be Aegon Targaryen, his family and their dragons, that fled west, and later conquered Westeros. So, since the Doom, the Targaryens had a monopoly on the world's dragons. Which leads to the second reason:

2. The Dance of the Dragons:

The Dance was an extremely messy and bloody war of succession between the Targaryens, and far beyond my ability to write on a short answer. What matters is that after the Dance most dragons were dead. Afterwards, when Aegor III took the throne, there were only 4 dragons alive, and they all died during his reign.

Answer 4

What happened with Daenerys' dragons eggs is actually a mystery, but it's generally accepted that the hatching was a consequence of something, and no one in Essos or Westeros knows that. If someone attempts to hatch another dragon egg, maybe they will not be successful, because they don't know what it takes.

In A Game of Thrones, Bran has a dream where he looked at Asshai and saw dragons stirring beneath the sunrise. Maybe there are dragons in the Far East that we don't hear about yet.

Answer 5

The most likely reason that there are no other dragons is simply plot armor for Daenerys. We are led to believe that she is "the one" from this miracle, even though the last Westerosi dragon laid 5 eggs. Also, Targaryens hatched dragons for hundreds of years; this isn't something out of the ordinary, it was just rendered difficult after the last dragons were weakened from generations in the Dragon Pit.