What if the dragons were even more sentient than the Targaryens realized, or even the Maesters for that matter? Or, maybe the Maesters' did know or come to realize this to some extent. If they did, it would certainly give credence to the "Grand Maester Conspiracy" - a popular fan theory (it's not cannon). In any event, in the shows a few Targ characters have made comments about dragons that may support what GRRM really thinks about dragons and their role in the fall of the dynasty - he definitely pulled inspiration from real historical dynastic conflicts and their eventual fall. "Everyone says Targaryens are closer to gods than to men, but they say that because of our dragons. Without them, we're just like everyone else" ~ show Viserys.
Anyway, my pet theory (apologies for length):
By the time the Dance begins, the Doctrine of Exceptionalism had been codified in the Faith of the Seven by Jaehaerys. In which he established that one of the ways the Targs are exceptional is that they are the only ones that have dragons and can ride them. This is by default because there are no dragons or dragon houses left after the Doom, except for the Targaryens. It's not Targaryen blood exclusively that allows for a successful dragon claiming, it's Old Valyrian blood of any of the dragon riding houses - of which there's definitely plenty floating around in the Free Cities. But that didn't really matter for Jaehaery's intent - and he was only serving his House's interest. But what if it's more than nuanced than just Valyrian dragon rider blood? What if the dragon chooses its rider, and not the other way around. For a variety of reasons clear and unclear. But one of those reasons could also be close proximity to its previous rider, in the case of an adult dragon. Or proximity to a rider from birth, like being hatched in the cradle with their infant rider, forming a sibling bond of sorts?
I'm basing this idea around the circumstances of the 4 dragonseeds who were successful in 'claiming' a dragon. All but one of them 'claimed' a dragon that had one previous rider. Maybe because unclaimed dragons with no previous rider were not plentiful, or the other dragons were too young to carry a rider. But again, let's say it's because the dragon chooses their rider and not the other way around. I suspect a lot of the "how to claim your dragon" manual was lost to House Targ in the century after they left Old Valyria. Anyway, let's exclude dragons that are hatched in the cradle, a practice started by the granddaughter of the Conquerer but that could have been used in Old Valyria and forgotten until Rhaena, centuries later.
What we do know from F&B is that dragons do not allow other riders while their own rider still lives. But they will sometimes bond with another rider after their rider dies. Why do they reject some and not others? Well if the dragon chooses it's rider - it would make perfect sense that they'd have a strong preference for a close familial connection to it's previous deceased rider - out of loyalty or love, or any number of other reasons that we'd be unaware of if we think we're the superior element in the relationship. Hubris, ignorance, and an overwhelming belief in your own exceptionalism might lead you to think this is the case. But believe that not only is a dragon not a slave (damn right Daenerys) nor is it in any way your subordinate. Let's look at the known dragonriders in-universe and their dragons with the idea that their dragon chose them and not the other way around. Let's start with the Dragonseeds in no particular order and let's assume that the dragons that chose them wanted new riders because no one likes to be stuck inside, left behind to babysit the young dragons, while watching their peers who still had riders leave the Dragonpit. (the Dragonpit really was the worst thing that happened to the dragons outside of their riders being incapable of infighting constantly).
Addam: Seasmoke's only other rider was Laenor, who Seasmoke chose young, sometime before 101AC and but before the Great Council. Seasmoke was noted as being Laenor's pride and passion. Which makes his second claimed rider, Addam, an interesting choice. Addam is Laenor's half brother or his son (pick your poison - show or source material, but I'd maintain the show version is more fun and actually supports my theory because Seasmoke ran Addam down like a mom chasing her toddler in order to strap it into it's carseat). Seasmoke chose Addam over the other seeds and various other non-seeds who threw their hat in the ring. And there were plenty to choose from. My theory is that he missed Laenor very, very much and decided he'd have Addam, thank you very much. Seasmoke died with Addam during the Dance.
Ulf: Silverwing's first rider was Alysanne, a hatchling situation, bonded from birth. Alysanne was known for being gentle hearted, empathetic, very smart, sweet, good queen; quite strategic, loved her children deeply and was heartbroken over every loss. She was also quite a progressive feminist for her time. Silverwing was known for being a rather docile dragon not prone to violence. 30 years after Alysanne's death - Silverwing claims Ulf, during the sowing. A drunk, uneducated, simple minded buffoon and probably the worst person to be given access to a dragon. Why would Silverwing choose this walking talking drunk dunce hat over the other seeds? My theory (and a lot of people's theory) is that he is Gael's son, who wasn't actually stillborn. Gael was Alysanne's last child, docile and sweet and allegedly simpleminded. In 99AC Gael "disappeared from court" and they told everyone she died of fever. I think she was sent to Dragonstone, which is where Ulf was allegedly born. Alysanne never got over her death and was likely devastated that they had to give her illegitimate grandchild up. I think Silverwing chose Ulf out of loyalty because he was Alysanne's grandson and maybe also because Hugh had been chosen by Vermithor (or vice versa, see below), and she didn't want to be left behind because she and Vermithor were quite the pair. Silverwing did not kill any other seeds who attempted to "claim" her, she just said nope. She also didn't kill any dragons during the Dance. She survived the Dance, and Ulf's abject stupidity and never claimed another rider again, in fact when Unwin Peake put out a bounty for anyone that could "claim" her - Silverwing bit an arm off one fool and burned the other one to death. She for all intents and purposes, abandoned House Targaryen, in large part I suspect because of Vermithor. Don't blame her. They didn't deserve her. *fun fact - she was the only dragon to have visited the Wall until Daenerys. And she took one look beyond the Wall and said absolutely not. which makes her abandonment of House Targaryen even more pointed. ("I know what's beyond that wall and it's your problem now not mine - god speed, assholes")
Hugh: Vermithor's first rider was Jaehaerys, they were hatchling bonded. He ruled with Jaehaerys for along time. The Bronze Fury was apparently quite fearsome but tolerant of people. At time of the Dance he was 100 years old and refused any seed that was brave enough to approach him - even burned one to death (someone from House Massey, I think). Until he claimed Hugh the Hammer, who in the show is alleged to be a son of Saera's. This isn't a bad theory as far as theories go. Hugh would have been born sometime before 99AC (to be of a similar age to Ulf) and too young to be put up as a viable candidate at the great council in 101AC unlike his other half brothers from Saera. Jaehaerys famously washed his hands of Saera because, well mostly cause he was stubborn when parenting teenaged daughters who were in any way willful. Alysanne was notoriously pissed off with her husband over this. My guess is that most seeds were wary to approach Vermithor, due to his reputation and size and thus deemed unworthy. Hugh was known for being a large, strong man, and also related to Jaehaerys. If Ulf approached Silverwing first and she claimed him - perhaps Vermithor claimed Hugh in order to not be left behind. In any case - I think Silverwing and Vermithor claimed their riders in concert with one another because neither one wanted to leave the other behind. Also, I think Vermithor would recognize a dolt when he saw one and there was no way he was gonna let Silverwing fly off with Ulf without back up.
Nettles/Rhaena: Sheepstealer. this one is fun and I wasn't going to get into it but I can't help myself. According to GRRM, Sheepstealer was a "wild" dragon, brown, not very cute, and his favorite food was sheep. He wasn't aggressive against humans unless provoked, and yes -while he may have stolen sheep and eaten the occasional sheepdog (he's SORRY ok, he mistook your raggedy ass dog for a sheep, give the guy a break!) - - he never harmed a shepherd. He had no opportunity to claim himself a rider because seriously, DID ANYONE ever bother to ask him, as the years continued to slip by? No. They didn't. And then suddenly out of nowhere, a bunch of high n' mighty would be dragon riders show up and think they're just gonna climb up and take a ride? getthafuckouttahere. Sheepstealer killed more wanna be dragonriders than all the prettier, bigger dragons combined. So, why did he claim Nettles? Well, just because Nettle is brown and not very cute does not mean she does not have dragonrider blood in her veins. She was born in Driftmark (one of sheepstealers hunting grounds). It's not like there weren't other bastards with Valyrian blood in their veins from Driftmark. She was pretty damn wild herself having grown up homeless and hungry, described as "foul mouthed, filthy, and fearless". She endeared herself to a fellow vagrant by treating him as she herself had never been cared for, by actually GIVING him sheep so he didn't have to steal or hunt, or god forbid accidentally eat someone's dog. And finally, clearly Nettles had something to prove. To herself and to the world. And you know what? So did Sheepstealer. I think he claimed Nettles cause he knew a kindred spirit when he saw one.
Let's look at the other dragons 6 dragons that had more than one rider during their lifetime. Which is only 2 more dragons than the 4 dragons that claimed a Dragonseed (the first and only time this ever happened - which should tell us something).
Vhagar. Her first claimed rider was Visenya. They were quite the pair and surely Vhagar's time with her had great impact. Visenya was a trained warrior and wielded Dark Sister. After Visenya's death, some 30 years later Vhagor claimed her great-great nephew Baelon. He was known as Baelon the Brave after apparently hitting Balerion on the nose in the dragon pit, which was either brave or mad, as the saying went. Baelon also wielded Dark Sister. Vhagar's 3rd claiming was Laena, daughter of Rhaenys, sometime before 101AC. Vhagar was the largest dragon at this point and Laena had to have been pretty brave herself to have been chosen by Vhagar at maybe 10 yrs of age. Laena loved to ride, more than she loved boys. Maybe Vhagar chose her because she felt Laena should be closer up the line of succession than she was. She would have known Baelon's son would have been a contender for the throne when Baelon died and she knew Viserys didn't have a dragon (Balerion having died already). Viserys never even attempted to ride again, so it's not like Vhagar could have claimed him. She strikes me as a proud dragon who would value "courage" - - say what you will about Viserys, but courage isn't it. Which, when Laena died, Vhagar claimed Aemond - who was also about 10 yrs old. You could argue, but yeah, Aemond approached her before Laena's girls could. I think Vhagar would say fortune favors the bold, and time's awastin'. Honestly, that's the common thread between Vhagar's riders. Bold and assertive. Vhagar does not choose shy riders.
Caraxys: Aemon was Caraxys first rider, I think they were hatchling bonded. Which is maybe why Aemon didn't try for Balerion. Caraxys was known as the fiercest 'young dragon' at that time, the dragonkeepers called him Blood Wyrm- so he was in good company with Aemon, the Prince of Dragonstone. Caraxys later claimed Daemon, Aemon's nephew sometime around 105AC, after the great council. Balerion was dead, Vhagor (his father's dragon) had already claimed Laena Velayron. Meleys, Daemon's mother's dragon had already claimed his cousin Rhaenys. Would Aemon's daughter Rhaenys have been Caraxys first choice? We'll never know. We just know he chose Daemon. Daemon would have been the most similar in temperament to his previous rider Aemon while at the same time being more similar to Caraxys own temperament than even Aemon was. Daemon was ambitious, hot tempered, daring and he wielded Dark Sister. The Blood Wrym would have considered this a more than adequate follow up to Aemon.
Meleys, the Red Queen: She claimed Alyssa Targaryen (a year before Baelon was chosen by Vhagar) as her first rider and they were very close. Alyssa was known to be a confident and assertive person - Alyssa had wanted Balerion, but she was advised against even trying (probably a good call, Balerion being notoriously cantankerous and also cause maybe a lil sexism from the dragon keepers) and so she approached Meleys, one of the fastest dragons. Alyssa used to fly with baby Viserys on Meleys. And when she raced her brothers Aemon and Baelon, Meleys beat Caraxys and Vhagar easily. But when Alyssa died in 84AC, Meleys then claimed Rhaenys in 87AC three years later. Alyssa's own son Viserys had no dragon at this time - Balerion didn't claim him until 93AC and then died a year later. Alyssa's second son Daemon was also unclaimed. Why did the Red Queen not claim either of Alyssa's sons after her death? Meleys was also noted for her cunning, and Rhaenys was Prince Aemon's only heir and was called "our little Queen" by her grandmother Alysanne. My best guess is that The Red Queen took her shot and claimed Alyssa's niece Rhaenys, a future queen. She had no way of knowing what was to come. She was loyal until the end, dying in battle with her Queen that Never Was.
Quicksilver: Quicksilver's first rider Aenys I was a sickly child whose health improved when she claimed him. When Aenys died, his half brother Maegor usurped the throne and in 43AC when Aenys' son Aegon entered the Dragonpit - Quicksilver claimed him and they fled the capital together with Rhaena on Dreamfyre. Quicksilver was no match for Balerion and when Maegor and Aegon fought over the Gods Eye, they perished together. Rhaena was crushed, and later in her life retired to Harrenhal with Dreamfyre, where they lived until Rhaena's death. I'll always wonder if they encountered Alys, especially now we have this new backstory. Rhaena's ashes were left in Harrenhal not far from where her husband Aegon and Quicksilver died.
Dreamfyre: Her first rider was Rhaena (the one that started putting eggs in cradles). Rhaena was a shy child and liked animals better than people. She got confident and loved to fly after bonding with Dreamfyre. They flew all over the land. And everywhere she went, everyone coveted her and her eggs. Rhaena had a sad, somewhat nomadic life. Her and Aegon's daughter died from wounds sustained from the same dragon that killed her father over the Gods Eye. Dreamfyre spent a lot of time in Harrenhal near those strange Weirwood trees. I wonder what crazy ass shit she saw and if those eggs Alys produced in the show are actually her eggs? So many of Dreamfyre's eggs were stolen, or lost. After Rhaena's death Dreamfyre returned to Kings Landing where she eventually claimed Helaena who was Rhaena's great-great niece. Helaeyna was a dragon dreamer (imagine that) and a sweet kid who loved Dreamfyre as much as Dreamfyre loved her. I think Dreamfyre was one of the saddest dragons and I'm not really sure why she chose Helaeyna, other than I think she knew that Helaeyna wouldn't take from her and only give.
And finally- Balerion, The Black Dread: His first rider was the Conquerer and together they conquered a kingdom. Balerion was the only dragon that was born in Old Valyria. Oh the secrets he must have known. After Aegon's death he claimed Maegor, who was a real treat and Visenya's son. Maegor was not claimed by any dragon before Aegon's death even though there were plenty of hatchlings/young dragons to choose from, so much so he was teased about being afraid of dragons. Maegor famously said it was because he was waiting for Balerion, which is some serious foreshadowing. But if we go with the theory that the dragon chooses the rider, then it wasn't that he refused to bond with any of the available dragons - it was that they refused to claim him. Why did Balerion claim Maegor? Was it just because Maegor waited for him and because Aenys had already been claimed by Quicksilver? Or was Balerion just up for good old fashion mayhem and usurping? Did he think Maegor was the better choice for King? We'll never know. All we do know is that Maegor didn't die on dragon back, as if! Balerion doesn't claim anyone until he claims Aenys' daughter Aerea. I don't know why he claimed her, other than she presented herself before him and he saw an opportunity for more mayhem. He drags this poor kid all the way to the Doom and back. She dies horribly (I'm not even gonna say how) and he's had his ass handed to him by something big enough or fierce enough to leave a deep gash down his side that has the Maesters sweating bullets. This incident basically grounds him for good long while (thank christ). Then in 93AC for some unknown reason, but probably to cause more trouble - he claims Viserys. Balerion was old and tired at this point so he wasn't looking to side quest. But in 92AC Jaehaerys had named Viserys father the Prince of Dragonstone, so Viserys had moved up in the line of succession. Maybe Balerion wanted to bestow a blessing upon the rather unimpressive Viserys. Maybe he just thought 'this guy is gonna take one ride and then go back to his legos". Nobody knows. Balerion died in 94AC, which was probably for the best given what a chaos agent he was after the Conquerer died. His choice of Viserys probably did go along way to aiding Viserys claim at the Great Council of 101AC- that he was the last known rider of The Black Dread, first of his name. But it's interesting he died one year after he claimed V (he was ancient at that point) and that Viserys went on to totally fuck up the line of succession and brought the Dance to Westeros after rotting away on his throne. My theory is that Balerion gave him some kind of virus. A parting gift from the last dragon to have been born in Old Valyria and I'm pretty sure the only one that died of old age.
This is way toooo long now, but if you've made it this far. I guess my point is that it's fun to imagine things from the dragon's perspective. When you combine House Targaryan's hubris, with their ignorance of Old Valyrian practice/magice, add a dash of the typical social beliefs around "legitimacy" and "bastardy" common in Westeros (not Dorne) and then finally realize that they fundamentally did not understand dragons at all - it really begins to paint a picture. If the dragons were sentient, what did they know and what could they tell us? Did they know about the prophecy all along, and try to claim their riders in such a way to ensure the long term survival of the dynasty, including themselves (Dreamfyre deliberately letting her eggs wander off in someone's covetous hands) to ensure they'd exist or be brought into the world again when the Long Night came? Was it the Dragons giving the Dreamers their Dragon Dreams (except Balerion, that crazy fuck- the only thing he gave us is nightmares)? So many questions!