We'll be talking up-to-date SPOILERS for both the books and the show.
1. No one knows what really killed all of Daenerys' ancestors
As far as Targaryens go, Daenerys is pretty much all we have left. Many fans hold that royal blood also flows through the veins of Jon Snow royal lineage (and depending on who you ask, Tyrion as well), but Dany is the last confirmed member of the bloodline. After 300 years of ruling Westeros, the Targaryen family was all but wiped out by Robert's Rebellion and its aftermath -- and in Viserys' case, a short coronation with a very hot crown.
But it wasn't always that way. See, the Targaryens were once just a very small part of a large empire called Valyria. You might remember it from an episode in season five of the TV show, when the buddy comic duo that was Jorah Mormont and Tyrion Lannister floated through the smoky ruins on their way to see Khaleesi.
In a fantasy universe as old and storied as this one, there are bound to be plenty of mysterious ruins. Hell, some parts of the world beyond Westeros would probably give Skyrim a run for its money in the "abandoned dungeons" department. But these towering remains once belonged to a great civilization that perished under mysterious circumstances. This cataclysmic event has come to be known as The Doom of Valyria. Though the exact cause of the Doom is uncertain, pretty much everyone agrees that it was an earth-shattering catastrophe the likes the world had never seen. As described in The World of Ice and Fire:
(...) every hill for five hundred miles split asunder to fill the air with ash and smoke and fire so hot and hungry that even the dragons in the sky were engulfed and consumed. Great rents opened in the earth, swallowing palaces, temples and entire towns. Lakes boiled or turned into acid, mountains burst, fiery fountains spewed molten rock a thousand feet into the air, and red clouds rained down dragongalss and the black blood of demons.
That last part with the demon blood rain miiight be an embellishment, but we have to remember that the information we have comes in the form of what's basically a legend. It's not exactly proven that say, an obsidian hailstorm pulverized the landscape, but the fact that something incredibly drastic happened is undeniable. Tyrion notes in A Clash of Kings: "I believe there were once dragons. I've seen their skulls, after all." Similarly, we've seen the bones of Old Valyria, and they're telling us a story in a language we can't understand.
Though it might be easy to say what destroyed Valyria was just a series of really, really big volcanic eruptions, it seems unlikely that we're looking at something so simple as a supercharged Pompeii situation. To understand what we're really dealing with here, we should go back and revisit Valyria as it was before. Back when it looked like this:
via Ted Nasmith
Around the time that Westeros had just finished dealing with the White Walkers and their horrifying ice spiders, Valyria was just getting on its feet. A comfortable distance from the tyrannical slavery empire that was Old Ghis, the Valyrians built themselves up in large part thanks to more than a dozen active volcanos that dotted the land. These massive mountains became known as the Fourteen Flames, and they were continually exploited for the wealth of precious metals dwelling inside them.
Oh yeah, and also Valyrians got themselves some dragons. That's pretty important.
Wielding that kind of immense power, Valyria struck down Old Ghis without breaking a sweat, and went to build a Rome-like empire for themselves. To be clear, this wasn't exactly a utopia for all involved. Everyone conquered by these silver-haired supermodels had to ditch their old language in favor of High Valyrian. The empire also made liberal use of slaves, who were forced to mine the rich ores of the Fourteen Flames in what you might call a "lifetime appointment." It should be no surprise that these kinds of conditions inspired many of the slaves to escape to form the city of Braavos; same with the Andals, who made a pilgrimage to Westeros to escape religious persecution.
Life kind of sucked for everyone else, but if you were a Valryian, you were probably pretty satisfied about how things were going. The one exception to this rule was Aenar Targaryen. Upon hearing of a dark prophecy that came to his daughter in a dream, Aenar took his family and everything he owned and booked it to a little hideaway off the coast of Westeros. You know it as Dragonstone.
via Philip Straub
It would be hundreds of years before Stannis Baratheon would take command of the island and all the grammar within. For now, the Targaryens had escaped what they believed to be certain death. And then... nothing happened. For over a decade, the family stewed on that rock, probably wondering what the hell they were doing there. And then, as the prophecy foretold, came the Doom.
It's tough to overstate just how complete and total the destruction of Valyria was. Almost all of the dragonlords were killed in the event itself, and the remaining few were done away with by enemies shortly thereafter. The only people left in the entire world that had access to (and domain over) dragons were the Targaryens, on their own little island of Dragonstone. As for Valyria itself... it's shattered beyond recognition, a piece of the world permanently broken. You can even spot its dessicated corpse on the world map, Southeast of Westeros..
via Jonathan Roberts
So what caused the Doom? While we may never get a clear answer, there are several theories floating around. The simplest explanation is that it was just a giant natural disaster, a mega-volcanic eruption a la Mt. Vesuvius. Some suggest that Valyria may have inadvertantly been at fault, that their unrepentant mining of the Fourteen Flames made the land less stable. Barring nature and/or hubris, there are others who might have wiped out the Valyrians on purpose.
In A Feast for Crows, Arya has a pretty interesting chat with a certain No One about the history of the Faceless Men:
"Revolts were common in the mines, but few accomplished much. The dragonlords of the old Freehold were strong in sorcery, and lesser men defied them at their peril. The first Faceless Man was one who did." "Who was he?" Arya blurted, before she stopped to think. "No one," he answered.
That very night he chose the most wretched of the slaves, the one who had prayed most earnestly for release, and freed him from his bondage. The first gift had been given." Arya drew back from him. "He killed the slave?" That did not sound right. "He should have killed the masters!" "He would bring the gift to them as well... but that is a tale for another day, one best shared with no one."
As it turns out, the dark assassins known as the Faceless Men were born with a grudge against Valyrians -- they started in the mines of the Fourteen Flames. Sure, they eventually settled in Braavos to worship the Many-Faced God, but mass enslavement/murder isn't the thing you can forget by unfriending someone on Facebook. Keeping that in mind, it would make a lot of sense if the Faceless Men were behind the Doom. They wouldn't even have to whip up an earth-shattering spell -- something like well-coordinated acts of sabotage would be more their speed. Attacking key Valyrian scientists and lords might have been enough to disrupt the emprie long enough for them to destroy themselves.
That said, it's still impossible to know what actually went down and if anyone was involved beyond a huge wall of hot lava. Knowing George R.R. Martin, maybe some maester started to write a book about it and never finished it.
2. The secret dragon eggs hiding under Winterfell
For most, the biggest mystery surrounding the Starks has long been spoiled; though not outright confirmed in the books or show as of this writing, "R+L = J" is less of a theory and more of a law of the land. Sort of like how kids who played Oddjob in Goldeneye were more likely to grow up to be sociopaths -- there's no scientific evidence, but everyone accepts it as fact. Much murkier, however, is the existence of dragon eggs beneath Winterfell. It sounds like a tinfoil hat conspiracy worthy of Captain Reynolds Wrap himself, but there's too much smoke to ignore this fire.
The whole thing hinges around a dragon named Vermax, who did indeed visit Winterfell about 170 years before Bran was pushed from that window. Vermax's rider Jacaerys Valeryon made the trip to hash out an alliance with the Starks and some of the other Northmen, and it was during the treating that Vermax supposedly found a nice place to pop out one or more eggs.
Now, this all comes from the mouth of Mushroom, a dwarven court jester who worked for the Targaryens for many years. After his long service, Mushroom published a tell-all that included a lot of wild stories that have largely been dismissed as fabricated lunacy by the maesters of Westeros. But we as readers have been taught by George R.R. Martin to never underestimate someone because of their stature.
So where are these dragon eggs supposed to be, anyway? Well, if Mushroom is right, they're in the crypts of Winterfell.
In-universe historians speculate that Winterfell was founded where it was because of the local hot springs. That seems likely, since a steady supply of warm water is a pretty nice perk in a world where even sorcerers haven't been able to figure out indoor plumbing. The crypts of the city are right up against these hot springs, which makes it the most probable place a cold-hating dragon would pop a squat and pinch off a few eggs.
But maesters are quick to dismiss this scenario altogether. Not only are they skeptical of anything a jester says, but there's also a the sticky matter of biology. Again from the World of Ice and Fire:
We can dismiss Mushroom's claim in his Testimony that the dragon Vermax left a clutch of eggs somewhere in the depths of Winterfell's crypts, where the waters of the hot springs run close to the walls, while his rider treated with Cregan Stark at the start of the Dance of Dragons. As Archmaester Gyldayn notes in his fragmentary history, there is no record that Vermax ever laid a single egg, suggesting the dragon was male.
Well, that would kinda put the kibosh on the whole thing, wouldn't it? It's not as though dragon DNA was spliced with tree frogs and life uh, found a way to make them switch sexes. Except that's actually pretty close to the truth! In A Dance With Dragons, one of the last things Maester Aemon says to Sam concerns this very topic.
"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."
Aemon is actually talking about how the prophecies of old could have been genderless in their text, and that the title of "The Prince That Was Promised" may well belong to Daenerys Targeryen. But the important part for us is that this implies it was biologically possible for Vermax to lay those eggs somewhere next to the sullen stone statues of the Starks underneath Winterfell.
via Thomas Denmark
The World of Ice and Fire disagrees with this premise, saying those who believe in sex-changing dragons are mistaken. But we have to remember that even though it's something of an enyclopedia, TWoIaF is written from the perspective of a fictional person within Westeros, a narrator that is biased and therefore completely fallible. At this point, it's the word of one maester against another -- we can't be certain who's in the wrong, or even if neither are in the right.
The only thing we do know is that there's something going on down in those crypts. They supposedly predate Winterfell itself, so anything is possible. This mystery has fascinated several people within the books and the TV series. King Robert, Lady Dustin, Mance Rayder and more are all itching to get downstairs for various reasons. Jon Snow, on the other hand, is not in a big rush.
"Somehow I know I have to go down there, but I don't want to. I'm afraid of what might be waiting for me."
If L+R=J is indeed true and Jon Snow is a secret Targaryen, could it be that his own dragon egg is waiting for him to claim it?
Then again, it may be that this is all symbolism, that Jon doesn't want to visit the graves of Starks because he's not really one of them. The "dragon beneath Winterfell" could be a fancy way of saying that a Targaryen is hiding in plain sight. Hell, there's nothing standing of the way of both theories being true, that Jon Snow is a Targaryen and will eventually ride a badass ice dragon into battle and give Ramsay the worst case of shrinkage ever. Again, knowing George R.R. Martin, we probably won't be so lucky.