Spoilers: This is only a theory, so there is no guarantee that this is correct. It also only includes information up to the end of the TV series, though some information used is only present in the books.
One of the most popular theories about Game of Thrones concerns exactly who who Jon Snow's father is, a theory commonly known as R + L = J.
Any revelation that a character has Targaryen blood would drastically change what's happening in Westeros.
So what if... TYRION LANNISTER WAS A TARGARYEN?
The story goes that King Aerys, father of Daenerys, was infatuated with Tywin's wife, Joanna.
Following on from this, he "took certain liberties" during the bedding ceremony. Some fans believe Aerys, either by seduction or force, continued a relationship with Joanna.
This betrayal also explains Tywin's violent hatred towards Tyrion, and especially his complaint that "I cannot prove you are not mine".
Tyrion's physical description in the books is actually closer to Targaryan than Lannister, with white-blond hair and different-coloured eyes.
The different-coloured eyes is a particularly compelling detail, as it implies two different lines coming together, as well as one of them being very dark, like Dany's.
Tyrion's personality also shows signs of not being truly Lannister, as he is described as having "a morbid fascination with dragons", and a bookish inclination, much like Rhaegar Targaryen.
The most important reason for this theory is a prophecy given to Daenerys in the House of the Undying: "The dragon has three heads."
Additionally, a dragon must be "born of blood", and Jon, Dany, and Tyrion all had mothers who died giving birth to them.
And in the last moments of his life, Tywin says what he has suspected for many years, which fans have interpreted as the final confirmation that Tyrion is not truly a Lannister.
If both theories were true, this would make Jon the nephew of Daenerys and Tyrion, who are half-brother and sister.
And if Tyrion IS indeed a Targaryen, what impact does that have on the rest of the series?
It leaves open the option for him to unite with Daenerys, which would be a fascinating alliance. And if Jon is indeed also part Targaryen, it could mean a three-way alliance to bring the Targaryens back to throne.
It could also change the order of succession for the Targaryen throne, moving Tyrion after Jon, with Daenerys in third, assuming their nature as bastards is no impediment. Of course, being Westeros, it's most likely that whoever has the biggest army will succeed.
Or the most dragons.