Throughout the show, the most we get to hear about the Targaryens is through the stories about the Mad King Aerys. They get the short end of the stick when it comes to lore and get shoved aside in the annals of Westeros' history as another House in the wheel.
But all of that is just Baratheon propaganda. Don't believe everything you hear, sheeple! From an island no bigger than King's Landing itself their legacy begins and they went on to conquer an entire continent, astride their legendary beasts.
The Targaryens were actually the greatest of the Great Houses of Westeros and here's why.
1. Aegon Targaryen I is the man who created the kingdom of Westeros.
Aegon wasn't just a conqueror, he was the man who united an entire continent of warring Kings under him.
Think about that for a second. The entire world of game of thrones exists today because Aegon had foresight. Instead of indiscriminate killing, he allowed kings to surrender and join with him. When they didn't he destroyed them. And he rode the biggest motherfucking dragon that has ever existed, Balerion the Black Dread.
2. The greatest warrior heroines of Westeros have all come from House Targaryen.
Visenya and Rhaenys Targaryen, the sisters-wives of Aegon and Rhaenyra Targaryen, one half of the Dance of the Dragons.
With the exception of Nymeria the warrior queen, there has scarcely been anyone who could compare with these three. Visenya and Rhaenys rode their dragons beside Aegon when they conquered the Seven Kingdoms. Rhaenyra literally said "Fuck You" to generations of primogeniture when she went to war against her own brother and father for the throne.
3. The Red Keep is the equivalent of the Taj Mahel in ASoIaF and it was built by a Targaryen.
Maegor Targaryen, the Cruel is remembered for his tyranny. But if the Targaryens were Mughals then he was Shah Jahan.
It was said that Maegor had every one who laboured on the Red Keep put to the sword so nothing of it's magnificence could ever be built again. Not only that but he also had multiple wives. In fact, he went to war against the faith and his own family when they objected his polygamy. And won.
4. The Targaryens were better friends of the Faith of the Seven than any other house.
Baelor Targaryen I, the Blessed was the one who built the Great Sept of Baelor in King's Landing, the center of all religious worship.
The Great Sept of Baelor is the greatest monument ever built to The Seven. It is the center of worship not just in King's Landing but all of Westeros. It was Baelor I who married the crown to the faith and he was known as Baelor the Beloved for he loved and was loved by the common folk. A truly benevolent king.
5. Targaryens were the ones who brought the feisty Dornish into the realm.
It was Baelor I who first forged the peace with Dorne and his son Dareon Targaryen II who brought them into the realm.
The Dornish trace their legacy to Nymeria, the warrior queen who came to Westeros while fleeing the dragonlords in Essos. Considering their ancient enmity, Baelor I's peace with the Dornish was no small feat, considering it was right after the Young Dragon's war of conquest on Dorne. It was his son, Dareon who cemented the alliance through marriage and so all of Westeros was united.
6. All the legendary swords in Westeros are forged with Valyrian steel, which the Targaryens brought to Westeros.
Valyrian steel is said to be forged with a special technique which involved using magic.
All the greatest swords of legend were those forged from Valyrian steel. They don't rust, they don't lose their edge and they're forged with magic. Unless Jon Snow cutting down the White Walker with Longclaw was a possible hint at something else (coughazorahaicough), we can be sure that these blades truly do have some mystical properties. And it was the dragonlords who brought the secrets of Valyria with them.
7. The Targaryens administered a bigger Kingdom than any other house, even the Starks, and they were bloody good at it.
They faced more rebellions than any other house combined and it was under Jaehaerys Targaryen that Westeros saw it's longest era of peace.
A lot of people seem to forget this little fact. Within two decades of their fall, Westeros has already plunged into complete chaos. That is the kind of poltically unstable kingdom that the Targaryens first forged and then successfully ruled for three centuries. With a few notable exceptions, as there always are, their reign was one of stability and even prosperity.
8. Some of the greatest heroes and rulers of the last three centuries came from House Targaryen.
Rhaegar the Dragonknight. Baelor II Breakspear. Aegon the Conqueror. Maegor the Cruel. Baelor the Blessed. Jaehaerys the Wise. Aegon V the Unlikely.
These are names right off the top of my head. That's not even mentioning the three badass heroines (but they deserved a point of their own). The Targaryens historically produced some of the greatest individuals to have lived in Westeros.
For that matter, even Maester Aemon was an exceptional man.
Three times he had the choice to cast away his oath and go where his heart willed and three times he put his duty before himself. He became a maester and then took the black just so that he couldn't be used as a tool in politics against his family.
9. Aerys Targaryen wasn't always a maniac but a man driven insane by tragedy.
The King who ruined their legacy forever wasn't just a madman, he was a tragic hero who was turned into a monster by life.
The mad king wasn't always mad. He was driven to insanity by successive stillbirths and his jealousy towards Tywin Lannister who, lets face it, was "a cunt." While the earlier part of his reign saw stability and even progress by the end he was delusional, paranoid and insane to the point that he could only get aroused by watching people burn to their deaths. His life, his rise and his fall were tragic.
10. Daenarys Targaryen has singularly had more badass moments than all other characters combined.
Even though she can be a whiny brat now and then, Daenarys is possibly the best suited contender for the Iron Throne.
In spite of all her shortcomings, Daenarys is still the only contender for the throne who still has any scruples or ideals. Her childish behavior does get tiresome after a while, but when you think about it her story started with her getting sold and then raped. From there she rose to be a queen and the mother of dragons.
The Targaryens are more than just the backdrop to a story. They are the ones who wove the entire tale, the fantastical world we entered into at the dim halls of Winterfell.
And if you're still not convinced then just remember this one thing. They rode on the backs of their goddamn dragons. That alone makes them win by default.