One of my favorite things to look into is “who all is going to ride a dragon by the end of the series?”. It gets discussed a ton, but its usually more of a how question than a why. I think there are some limiting factors (blood, reasoning, power), as well as some helpful factors as well (dragonhorn, powerful sorcery). In this post my goal is to look at more of the "why" than the "how".

List of Potential Dragonriders in the Series

Note: I don’t think all of these characters will end up riding dragons. But obviously several of them will (in addition to Daenerys).

Confirmed Dragonriders

Daenerys Targaryen

If you don’t know that Daenerys Targaryen is a dragonrider yet, I don’t know why you are reading this post. Daenerys Targaryen will ride Drogo (aka lil Balerion) and likely burn at least once city. Her invasion will mirror her ancestor, Aegon I's.

She bonds with Drogo as of the end of ADWD:

The dragonlords of old Valyria had controlled their mounts with binding spells and sorcerous horns. Daenerys made do with a word and a whip. Mounted on the dragon's back, she oft felt as if she were learning to ride all over again. When she whipped her silver mare on her right flank the mare went left, for a horse's first instinct is to flee from danger. When she laid the whip across Drogon's right side he veered right, for a dragon's first instinct is always to attack. Sometimes it did not seem to matter where she struck him, though; sometimes he went where he would and took her with him. Neither whip nor words could turn Drogon if he did not wish to be turned. The whip annoyed him more than it hurt him, she had come to see; his scales had grown harder than horn.

And no matter how far the dragon flew each day, come nightfall some instinct drew him home to Dragonstone. His home, not mine. Her home was back in Meereen, with her husband and her lover. That was where she belonged, surely. -ADWD, Daenerys X

She could start burning stuff as soon as her first or second TWOW chapter:

She flew across the Dothraki sea, high and higher, the green rippling beneath, and all that lived and breathed fled in terror from the shadow of her wings. -AGOT, Daenerys IX

Other "heads" of the Dragon

The purpose of the other "heads" of the dragon (even though one doesn't "necessarily have to be a Targaryen) would be to aid Dany (at least at first)

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. -ASOS, Daenerys VI

Jon Snow

Jon Snow as the son of Rhaegar/Lyanna would be a "song of ice and fire".

- Jon is mentioned with ice dragons numerous times (the constellation and the wall tunnels, etc.)

- What better way to show the realm that he has valyrian blood than to mount a dragon

fAegon VI Targaryen/Blackfyre

Whether you believe fAegon to be real or not, you do need to have actual dragons on both sides for a true "dance". And while its possible that GRRM has changed his stance on having one, he has mentioned it, and our favorite chubby little girl Teora has this vision:

It was then that pasty, pudgy Teora raised her eyes from the creamcakes on her plate. "It is dragons."

"Dragons?" said her mother. "Teora, don't be mad."

"I'm not. They're coming."

"How could you possibly know that?" her sister asked, with a note of scorn in her voice. "One of your little dreams?"

Teora gave a tiny nod, chin trembling. "They were dancing. In my dream. And everywhere the dragons danced the people died." -TWOW, Arianne I

and Mel sees this:

Through curtains of fire great winged shadows wheeled against a hard blue sky. -ADWD, Melisandre I

The point of Jon/fAegon riding dragons would be to aid Dany and both possibly betray her similar to the previous Dance with Ulf the White/Hugh the Hammer and then Addam Velaryon (proud to be faithful).

. . three heads has the dragon . . . the ghost chorus yammered inside her skull with never a lip moving, never a breath stirring the still blue air. . . . mother of dragons . . . child of storm . . . The whispers became a swirling song. . . . three fires must you light . . . one for life and one for death and one to love . . . Her own heart was beating in unison to the one that floated before her, blue and corrupt . . . three mounts must you ride . . . one to bed and one to dread and one to love . . . The voices were growing louder, she realized, and it seemed her heart was slowing, and even her breath. . . . three treasons will you know . . . once for blood and once for gold and once for love . . . -ACOK, Daenerys IV

Enemies of the Dragon

While its possible that Jon/fAegon could end up as enemies, IF any of this category somehow obtain a dragon (or other beast depending on what Euron is summoning), it will like be as an antagonist to Dany's plotline. Especially since of the visions of The House of the Undying seems to imply "three false dragons" that she will need to slay:

Glowing like sunset, a red sword was raised in the hand of a blue-eyed king who cast no shadow (Stannis as false AA). A cloth dragon swayed on poles amidst a cheering crowd.(fAegon as the false king of Westeros/Blackfyre) From a smoking tower, a great stone beast took wing, breathing shadow fire. (Euron as false god) . . . mother of dragons, slayer of lies . . . -ACOK, Daenerys IV

Euron Greyjoy

The Forsaken is full of imagery of Euron's potential summoning/attack on Oldtown. Euron's goal is to bring on the apocalypse/become a god. His summoning could work, he could get a dragon from the horn/Victarion as well.

"Only their shadows," Moqorro said. "One most of all. A tall and twisted thing with one black eye and ten long arms, sailing on a sea of blood."-ADWD, Tyrion VIII

Euron also desires to marry Dany as well. While I think there are other better options, the visions in the Forsaken could indicate this. He could betray her as well.

Stannis Baratheon

Stannis at some point is going to burn Shireen in order to "wake the stone dragon". There are numerous theories how exactly Stannis gets said stone dragon (I don't really agree) but they include: Cannibal/Summer's Winged Snake/etc.)

Stannis as the Night's King 2.0 would allow for this as well.

Stannis did not pull away from Melisandre's touch as he had from his queen's. The red woman was all Selyse was not; young, full-bodied, and strangely beautiful, with her heart-shaped face, coppery hair, and unearthly red eyes. "It would be a wondrous thing to see stone come to life," he admitted, grudging. "And to mount a dragon . . . I remember the first time my father took me to court, Robert had to hold my hand. I could not have been older than four, which would have made him five or six. We agreed afterward that the king had been as noble as the dragons were fearsome." Stannis snorted. "Years later, our father told us that Aerys had cut himself on the throne that morning, so his Hand had taken his place. It was Tywin Lannister who'd so impressed us." His fingers touched the surface of the table, tracing a path lightly across the varnished hills. "Robert took the skulls down when he donned the crown, but he could not bear to have them destroyed. Dragon wings over Westeros . . . there would be such a . . ." -ASOS, Davos V

It should be noted that unlike Euron, Stannis does have Valyrian blood (Egg's daughter married into House Baratheon).

That night she dreamt that 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. -ASOS, Daenerys III

Vessels

This category is for characters whose only reason to ride a dragon would be to get it from one place to another or die trying.

Victarion Greyjoy

As Victarion Greyjoy doesn't have valyrian blood, he must rely on the dragonhorn:

Bitterly Victarion brooded on the treachery of brothers. Euron's gifts are always poisoned. "The Crow's Eye swore this horn would bind dragons to my will. But how will that serve me if the price is death?"

"Your brother did not sound the horn himself. Nor must you." Moqorro pointed to the band of steel. "Here. 'Blood for fire, fire for blood.' Who blows the hellhorn matters not. The dragons will come to the horn's master. You must claim the horn. With blood."

and:

By the time Plumm and his companions came galloping back from the camp of the Girl General, the white dragon had flown back to its lair above Meereen. The green still prowled, soaring in wide circles above the city and the bay on great green wings. -TWOW, Tyrion I

It also should be noted that the Greyjoys owe Bittersteel and by default fAegon a blood debt.

Brown Ben Plumm

As we have seen in the series, Brown Ben potentially has two drops of Valyrian Blood.

Her captains bowed and left her with her handmaids and her dragons. But as Brown Ben was leaving, Viserion spread his pale white wings and flapped lazily at his head. One of the wings buffeted the sellsword in his face. The white dragon landed awkwardly with one foot on the man's head and one on his shoulder, shrieked, and flew off again. "He likes you, Ben," said Dany.

"And well he might." Brown Ben laughed. "I have me a drop of the dragon blood myself, you know." - ASOS, Daenerys V

Both Victarion/Brown Ben if they end up as dragonriders, are most likely a means to an end. They show the reader that you don't need to be a trueborn Targaryen to ride a dragon (Brown Ben), and Victarion's horn (if it works) could either going to marry Dany and go back to Westeros with her and then die or go back on his own and "provide" a dragon to Euron or fAegon.

The same can be said in the even unlikelier cases such as Moqorro/Tattered Prince.

Wildcard

The only reason I am including this one is because of all the imagery in the book already.

Bran/Tyrion

Tyrion is constantly used as an info dump about dragons and there are numerous passages associating him with dragonriding/special saddles/etc.. The only way I see this happening is through Bran warging or the slim possibility of Tyrion Targaryen.

When he was still a lonely child in the depths of Casterly Rock, he oft rode dragons through the nights, pretending he was some lost Targaryen princeling, or a Valyrian dragonlord soaring high o'er fields and mountains. Once, when his uncles asked him what gift he wanted for his nameday, he begged them for a dragon. "It wouldn't need to be a big one. It could be little, like I am." His uncle Gerion thought that was the funniest thing he had ever heard, but his uncle Tygett said, "The last dragon died a century ago, lad." That had seemed so monstrously unfair that the boy had cried himself to sleep that night.

I admit I do find this pretty unlikely but if you are interested: Tyrion the Dragonrider.

It should also be noted that not only does Bloodraven have valyrian blood (he's in a tree keep in mind), but he also possessed a dragon egg.

Failures

Not all men are meant to be dragonslayers. As with any character without Valyrian blood, we have also seen characters with valyrian blood die.

Dragons are not horses. They do not easily accept men upon their backs, and when angered or threatened, they attack. Sixteen men lost their lives during an attempt to become dragonriders. Three times that number were burned or maimed. Steffon Darklyn was burned to death whilst attempting to mount the dragon Seasmoke. Lord Gormon Massey suffered the same fate when approaching Vermithor. A man called Silver Denys, whose hair and eyes lent credence to his claim to be a bastard son of King Maegor the Cruel, had an arm torn off by Sheepstealer. As his sons struggled to staunch the wound, the Cannibal descended on them, drove off Sheepstealer, and devoured father and sons alike. -The Princess and the Queen

and:

Even for a son of House Targaryen, there are always dangers in approaching a strange dragon, particularly an old, bad-tempered dragon who has recently lost her rider. His father and mother would never allow him to go near Vhagar, Aemond knew. So he made certain they did not know, sliding from his bed at dawn whilst they still slept and stealing down to the great outer yard where Vhagar and the other dragons were fed and stabled. The prince had hoped to mount Vhagar in secrecy, but as he crept up to the dragon a boy’s voice rang out. “You stay away from her!”

In the main series we have one so far:

Quentyn Martell

Someone will. Many someones, like as not. "Sit, my friend." When Belwas sat and crossed his arms, Ser Barristan went on. "Quentyn Martell died this morning, just before the dawn."

The Widower laughed. "The dragonrider." -ADWD, The Queen’s Hand

Quentyn thinks that his valyrian blood will help him ride a dragon, Quentyn's death shows that it is no guarantee (as we also see with with Alyn Velayron and possibly Silver Denys).

Other Thoughts

There are numerous other characters with valyrian blood currently alive in the series and since we know that every character (to the best of our knowledge, yes even Nettles) has Valyrian blood.

That said numerous of these characters don't seem to have a "reason" to ever ride a dragon (ex: Brienne, Arianne, Edric Storm).

This post could also be affected by the following things as well:

• Pre-Valyrian Dragons in Westeros as the three Wild Dragons might not be of the same lineage (which would make some sense wrt Nettles)
• How the riderless dragons could get to Westeros
• Several dragons were still alive after the first dance. We don't know of their fates, but they would be extremely old right now. Especially if say the Cannibal was hiding out on skagos.
• Dragon strength/how to kill a dragon
• Alys Rivers aka the Witch Queen of Harrenhal has a "dragon" after the dance. This is probably a glamour on Caraxes' corpse. This could be repeated in some way

I'd like to reiterate that I don't necessarily think all of the characters I mentioned will ride one. But I think its obvious that both Viserion and Rhaegal will each have at least 1 rider (and probably more ) each.

Please keep in mind that while dragons have more than one rider, no rider has ridden more than one dragon. As we see with Joffrey/Syrax, dragons bond with one rider (keep in mind how Viserion/Rhaegal basically are no longer Dany's now that she has bonded with Drogon).

"You … you mean to ride them?"

"One of them. All I know of dragons is what my brother told me when I was a girl, and some I read in books, but it is said that even Aegon the Conqueror never dared mount Vhagar or Meraxes, nor did his sisters ride Balerion the Black Dread. Dragons live longer than men, some for hundreds of years, so Balerion had other riders after Aegon died … but no rider ever flew two dragons."

Viserion hissed again. Smoke rose between his teeth, and deep down in his throat they could see gold fire churning. -ADWD, Daenerys VIII

TLDR: A quick list of the most likely dragonriders and the reason on the "why" they would ride a dragon.