Nagas are children of Sage Kashyapa and Kadru. Adisesha, elder among the Nagas, is bed for Lord Vishnu. Vasuki, second child, is Lord Shiva's snake. There are many other Nagas but these two Nagas are prominent.
Chapter 39: Story of JaratKaru of Astika Parva: Adi Parva of Mahabharatha describes Vasuki as Nagaraj and Nagendra which are words for "King of Nagas".
हितो ह्ययं सदाऽस्माकं प्रियाकारी च नागराट्।
प्रसादं कुरु देवेश शमयास्य मनोज्वरम्॥७॥
The king of the snakes is always our friend and benefactor; O lord of the gods, be gracious to him and remove the fever of his mind.
ब्रह्मोवाच मयैव तद्वितीर्णं वै वचनं मनसाऽमराः।
एलापत्रेण नागेन यदस्याभिहितं पुरा।।८।।
तत्करोत्वेष नागेन्द्रः प्राप्तकालं वचः स्वयम्।
विनशिष्यन्ति ये पापा न तु यै धर्मचारिणः॥९॥
Brahma said :
O immortals, I have thought in my mind what you have said. Let the king of the snakes do what Elapatra had told him before. When the time comes, the wicked only will be destroyed and not the virtuous.
But Chaper 43 Astika's bite says Takshaka as Nagendra, King of Snakes.
दश नागेन्द्र वृक्षं त्वं यद्येतदभिमन्यसे।
अहमेनं त्वया दष्टं जीवयिष्ये भुजंगमः॥ ३॥
O snake, if you are so minded, bite (the tree then). I shall revive it, though bitten by you.
Wiki says Hindu texts call Adisesha, Vasuki and Takshaka by this Title.
How post of "King" can be conferred to three snakes? Are there any stories regarding coronations of any of these Nagas as King?