Mad Targaryens

"Babies or no, theirs was the same cursed blood that flowed within the Mad King's veins. They were 'dragon-spawn', and couldn't be allowed to survive. What would they grow to be, loyal subjects?...Was it a crime to put an end to a family of lunatics born of incest?"

- Robert Baratheon In A Song of Ice and Fire novels and its TV series adaptation, the Mad Targaryens refers an extreme and villanious non-official faction within House Targaryens, given to the extremely psychopathic family members of House Targaryen, due to the marriage between its close reletives.

History
House Targaryen carries the trait for insanity in its bloodline. Over three centuries of heavy inbreeding, marrying brother to sister whenever possible to "keep the bloodline pure," resulted in many of the medical problems seen with incest, particularly mental instability.

The most prominent example of the Targaryen madness was the last Targaryen king, Aerys II, who subsequently became known as "the Mad King". Aerys suffered from some sort of paranoid schizophrenia, hearing voices in his head which weren't real and which told him to burn people alive because they were all plotting against him.

Other Targaryens displayed highly bizarre personalities, though they were not strictly speaking "insane". They didn't have hallucinations or hear voices, but some had aggressive, sociopathic personalities, such as Aerion Brightflame - older brother of Maester Aemon. Eventually, Aerion succumbed to the delusional belief that if he drank wildfire it would transform him into a dragon: he was wrong. The effects worsened with each generation of compounded inbreeding. By the later centuries of their rule, it was joked that when a new Targaryen was born the gods would flip a coin to decide if they would be a brilliant statesman or insane.

Known mad Targaryens

 * King Maegor the Cruel was perhaps the first mad Targaryen in Westros history. He usurped the throne from his nephew and promptly decapitated the one Archmaester who protested. As king, Maegor turned to brutal tactics to suppress the Faith of the Seven, even riding on his dragon Balerion to burn down a Sept with all worshipers inside, using archers to pick off stragglers. He was found dead seated on the Iron Throne, his wrists slashed by the blades.
 * King Baelor the Blessed was overzealously obsessed with religion and purity, to the point that he starved himself into an early grave because he believed that food is of this world, and the material world is sinful. However, even with his madness, Baelor himself was neither a corrupt king nor a tyrant, making him by far the best and the most pious of all the Targaryen kings. According to history, he starved himself to death, by the prolonged fasting to cleanse himself of lust, but some believe he was poisoned by his Hand and uncle, Viserys.
 * Prince Rhaegel, older brother of King Maekar, was meek, mad, and sickly. He had massive delusions and was known to randomly take off his clothes and dance naked through the halls of the Red Keep. However he was described as kind and gentle. He died whilst choking on a lamprey pie.
 * Prince Aerion Brightflame, son of King Maekar and nephew of the also-mad Prince Rhaegel. Aerion killed himself drinking wildfire, believing it would turn him into a dragon, instead he died screaming. A year later, after the death of his father, Aerion's infant son Prince Maegor was passed over by a Great Council in favor of Maekar's youngest son Aegon, partially out of fear that Maegor had inherited his father's madness.
 * King Aerys the Mad, nephew of Aerion. The infamous Mad King. Started as a brilliant youth until he slowly turned into a perverted paranoid man. Had violent visual and auditory hallucinations, as well as a near permament feeling of persecution and jealousy, which drove him to execute anyone who slighted him out of fear that they were conspiring against him. Tried to blow up King's Landing with jars of wildfire, believing he would rise from the ashes as a dragon... and it eventually killed him. Betrayed and slain by his own Kingsguard, Ser Jaime Lannister the Kingslayer.
 * Viserys, the Beggar King, second son of Aerys II, suffered from child trauma and grown into a broken bitter boy in denial. He was nice and caring to his sister, Daenerys, until they had to beg and sell their memories for survival. He had delusions of greatness and would have frequent, violent outbursts of jealous anger towards Daenerys. Died as an exile in Essos, at the Dothraki Sea, after being "crowned" with melted gold on his head by Khal Drogo.

Madness in House Blackfyre
"All the same, his descendants continued to threaten the Targaryens until half a century later, when Barristan the Bold slew Maelys the Monstrous on the Stepstones, extinguishing Daemon's line."

- Varys, referring to the Blackfyre rebellions after Daemon Blackfyre's death Members from the infamous House Blackfyre, which originated from Daemon Blackfyre, a bastard of Aegon IV Targaryen, were possibly affected by the madness with Targaryen blood, so that they rebelled four times to ursurp the Iron Throne. Maelys Blackfyre, known as "Maelys the Monstruous", who is the last Blackfyre ursurper, was also discribed as an extreme and unpleasant savage.

Trivia

 * Even though House Targaryen was filled with madness, savage and disgrace it brought, this does not mean that all Targaryens were as a rule mentally unstable, as seen with how  Daen erys Targaryen  is still mostly mentally stable while her own brother  Viserys  is highly disturbed. However, Daenerys can show signs of madness by killing horribly people who angered her, especially during her frustrating rule as Queen of Meereen, where she allows the Shavepate to torture rich families for being suspects of treason and abandons Meereen and Astapor to chaos and death, while thinking only of Daario Naharis's body, Her TV series counterpart is a bit more reasonable as less violent though. Daenerys's oldest brother  Rh aegar  was considered a great man by all who knew him, though him escaping with Lyanna Stark can be considered a sconsiderate action that only caused a war and the death of Rhaegar's family. Maeste r Aemon  became a wise and revered advisor to the  Ni ght's Watch  for many decades, until everyone forgot his true identity as a Targaryen. The cunning manipulating Targaryen bastard known as Brynden Rivers, or Lord Bloodraven (three-eyed crow or three-eyed raven), disappeared shortly after bringing Aemon to the Wall.
 * Daenerys is not thought to be insane, though her merciless treatment of slave masters in Meereen is a dangerous hint that she may not have escaped from the Targaryen madness. Even so, her actions were ultimately against a group who were mistreating others (seeing this as an echo of how she was mistreated), and she thought she was helping to secure freedom for the slaves of the region. She's trying to learn how to rule, yet she's failing at every step also due to her quick temper and easy lust for the manipulating and dangerous Daario.
 * However, Barristan Selmy notes that the Mad King also genuinely believed his actions were just, even though others knew he was a madman, which compels Daenerys to become more merciful.
 * She also became known by most cultures in Essos as 'the Mad King's Daughter', due to her destroying the economy and trying to destroy an ancient culture and replacing with her own one.