Ogygia

Ogygia (Ὠγυγίη or Ὠγυγία in Ancient Greek) is the island that Calypso inhabits.

Description
Ogygia (pronounced oh-jee-jee-ah) is a phantom island that exists everywhere and nowhere. It was designed as the prison for Calypso, the nymph and daughter of Atlas. It is a very gorgeous place, where there are invisible servants and a beautiful garden, made by Calypso herself. One of the garden's plants is moonlace. Time moves differently there.

When Percy is leaving he says that he could come back and visit sometime, but Calypso tells him that that "No man has ever found Ogygia twice."

The hero Odysseus was marooned on Calypso's island thousands of years ago for seven years, (although some sources say five, three or even one). He was saved by a plea from Athena to Zeus who sent a message to Calypso by Hermes to release him for his journey to Ithaca or suffer the consequences. Calypso finally, though reluctantly, allowed Odysseus to build a small raft, gave him food and wine, and let him depart the island.

The Battle of the Labyrinth
Hera sent Percy Jackson to Ogygia to help him heal after he caused Mount Saint Helens to erupt. He was there for awhile before waking up and he was cared for by Calypso. The island seemed like a paradise to Percy, which instantly made him suspicious, as in his experience paradises normally are some kind of trap. However, Calypso soon convinces him that she means him no harm as she plants new flowers in her garden. When he asked how long he had been there, Calypso has problems answering as time moves differently on her island (is it is a phantom island, separated from the outside world).

Percy and Calypso soon grow close to each other, talking about their respective worlds. Percy finds Calypso's island to be beautiful, with its' large gardens that Calypso planted and a good view of the night sky. Calypso on the other hand is fascinated by Percy's description of Manhattan, but is saddened by its' lack of gardens and that she could never leave her island, as Ogygia was also her prison. She was never allowed to leave the island as she supported her father during the first war. The gods do visit her from time to time (as one unnamed god visited during Percy's stay on the island), and they tell her about the outside world.

Soon, Hephaestus appeared on the shores of Ogygea, asking Percy if he was going to stay on the island or leave to save his friends, but he would wait until the following morning for an answer. Calypso offered Percy the chance to stay on her island, explaining the full details of her curse, but Percy couldn't stay and decided to go home. Calypso understood that he couldn't stay and offered him a way off the island. Calypso said to Percy "Plant a garden in Manhattan for me, will you?" When he left, he claimed that staying in Ogygia and being with Calypso would always be his biggest what if.