Queen Rhaella Targaryen the sister-wife of King Aerys II Targaryen, The Mad King.
Rhaegar Targaryen
What became of the dream of dragons was a grievous tragedy born in a moment of joy. In the fateful year 259 AC, the king summoned many of those closest to him to Summerhall, his favorite castle, there to celebrate the impending birth of his first great-grandchild, a boy later named Rhaegar, to his grandson Aerys and granddaughter Rhaella, the children of Prince Jaehaerys.
The World of Ice and Fire, The Targaryen Kings: Aegon V
This would have then made Rhaegar 41 in 300 AC.
Viserys Targaryen
His Grace's new fidelity was apparently pleasing to the Mother Above, it must be said, for the following year, Queen Rhaella gave the king the second son that he had prayed for. Prince Viserys, born in 276 AC, was small but robust, and as beautiful a child as King's Landing had ever seen. Though Prince Rhaegar at seventeen was everything that could be wanted in an heir apparent, all Westeros rejoiced to know that at last he had a brother, another Targaryen to secure the succession.
The World of Ice and Fire, The Targaryen Kings: Aerys II
This would have then made Viserys 24 in 300 AC.
Daenerys Targaryen
I wasn't able to find an explicit quote proving that Daenerys was born to Rhaella but seeing as Viserys never forgave her about her mothers death it implies she has the same mother.
She had been born on Dragonstone nine moons after their flight, while a raging summer storm threatened to rip the island fastness apart. They said that storm was terrible. The Targaryen fleet was smashed while it lay at anchor, and huge stone blocks were ripped from the parapets and sent hurtling into the wild waters of the narrow sea. Her mother had died birthing her, and for that her brother Viserys had never forgiven her.
A Game of Thrones, Daenerys I
For working out Daenery's age is ab bit complicated but I have an answer on it here which concludes that she was born in 284 AC. This would then make Daenerys 16 in 300 AC.
Disclaimer: 300 AC was picked only because it made the maths easier and no other reason than that.
As for why there are such large age gaps well Rhaella had a lot of trouble giving birth with some stillborns and some dying not long after being born. Also Aerys is said to have slept around a lot so his attention was not always on his wife which may have caused problems.
Sadly, the marriage between Aerys II Targaryen and his sister, Rhaella, was not as happy; though she turned a blind eye to most of the king's infidelities, the queen did not approve of his "turning my ladies into his whores." (Joanna Lannister was not the first lady to be dismissed abruptly from Her Grace's service, nor was she the last). Relations between the king and queen grew even more strained when Rhaella proved unable to give Aerys any further children. Miscarriages in 263 and 264 were followed by a stillborn daughter born in 267. Prince Daeron, born in 269, survived for only half a year. Then came another stillbirth in 270, another miscarriage in 271, and Prince Aegon, born two turns premature in 272, dead in 273.
The World of Ice and Fire, The Targaryen Kings: Aerys II
The march of the king's madness seemed to abate for a time in 274 AC, when Queen Rhaella gave birth to a son. So profound was His Grace's joy that it seemed to restore him to his old self once again...but Prince Jaehaerys died later that same year, plunging Aerys into despair. In his black rage, he decided the babe's wet nurse was to blame and had the woman beheaded. Not long after, in a change of heart, Aerys announced that Jaehaerys had been poisoned by his own mistress, the pretty young daughter of one of his household knights. The king had the girl and all her kin tortured to death. During the course of their torment, it is recorded, all confessed to the murder, though the details of their confessions were greatly at odds.
ibid