When viewed from the air, the Polynesian atoll of Nukutepipi looks like a curved index finger, seductively beckoning you to its shores. Over a decade ago, Guy Laliberté, the billionaire founder of Cirque du Soleil, took the proverbial bait and bought this approximately one-square-mile paradise in the Tuamotu Archipelago, complete with a protected lagoon and coral reefs. He’s spent the years since turning it into a private resort for up to 50 guests, available for full buyout at the rate of $1.12 million for a seven-night stay.

This resort, as well as Laliberté’s other property on the Big Island of Hawaii, was designed by French architect Richard Dulude and furnished in a relaxed yet surprisingly urban style, with some pieces from Dedon and Restoration Hardware. Laliberté’s two-bedroom master residence is nestled in a prime spot amid tropical vegetation, with a private pool and large outdoor lounge. There are two junior villas with two bedrooms, outdoor lounges, and private pools, and 13 individual bungalows line the unspoiled beach.

Laliberté intends to turn Nukutepipi into a completely self-sustainable, carbon-neutral operation—no small task when the closest proper city and port is about 450 miles away. They’ve planted 25 types of organic fruit trees, grow 37 types of organic vegetables, have bees and chickens, collect rainwater, and use solar energy to power generators. While the location can prove a logistical challenge in terms of construction and ferrying of goods and services, it also guarantees a level of privacy that is increasingly hard to find, making the resort a seemingly ideal destination for those who require maximum discretion; guests, even those with private planes, must charter a flight from Papeete to reach the island.

Those who delight in simple pleasures will bliss out at the views from every vantage point, from their bed to strategically placed hammocks along the beach. Various species of birds are encountered at every turn and hermit crabs race to the sea en masse at certain times of day. Thought-provoking sculptures by the likes of Jim Dine and Wim Delvoye are placed throughout the property.

One of Laliberté’s primary concerns was the number of available activities so that nobody would ever get bored. In addition to the expected water sports and pursuits, guests can hop on a bicycle or partake in tennis, basketball, volleyball, mini golf, or archery. They can go stargazing in an observatory, climb a staircase to an antique telescope for whale watching, work out in an expertly outfitted gym, indulge in a spa treatment, hit the game room for a round of high-stakes poker or perhaps some shisha while watching a movie in the cinema there, or make use of the professional sound system in the “Kaipoa” beach bar. For meals, there are two full-service restaurants and a food truck. Fitness coaches, yoga instructors, Polynesian musicians and dancers, and even tattoo artists can be brought in, at an additional cost. nukutepipi.com  —Rima Suqi