Ricky Martin has one off Angra dos Reis in Brazil, Johnny Depp has Little Halls Pond Cay in the Bahamas and I, for a few exhilarating, make-believe moments, have my own private island in the Fijian archipelago.
Let me describe Laucala. From the air, viewed from the cream-leather cocoon of a seven-seater Beechcraft, great meringues of cloud dissolve to reveal jungle-covered monoliths tumbling down to the sea. The shoreline is etched into sandy crescents shaded by palms and lapped by ice-blue water mottled with corals. Off the north-eastern flank a fringing reef creates a permanent surf break that acts like a metronome, setting the tempo of the island. (It is a very sluggish surf break.)
We land on a strip of red earth shorn from the forest. Beneath a palm-thatched shelter, an assembly of Fijians with flowers behind their ears break into a song of welcome. The words are meaningless to me but their big bula smiles convey the sentiment.
Staff escort me to a shiny Land Rover and we drive through primeval forest - Jurassic Park, they call this tangled corridor of green - before coming upon tamer, more hospitable hinterland. Our brief journey ends on a plateau, at the thatched porch of something called Yanuyanu.
"It means 'the islands'," says Nilesh Kumar, the concierge assigned to greet me. "From here you can see the islands of Yavu, Yanuca and Maqeaa on the horizon. Come." He leads the way through bamboo gates to a perfumed garden and a compound of bures perched on a bluff above a scene of such shimmering loveliness it is impossible to re-create with words. My eyes pinball from the infinity pool studded with blue, green, black and white beach pebbles to the chic pavilion with alfresco lounge and dining, to the ginger and heliconia-edged sun terrace that falls away to forested hillside. Below are lush lawns and the immaculate grid of a palm plantation extending towards a shoreline of biscuit-coloured sand and, beyond, the sparkling South Pacific. And yes, there on the far horizon I can see three little silhouettes of the yanuyanu, though this gorgeous villa could as easily have been named after the open sea, the jungle or the waves, stars, tides, butterflies… except those words have already been used to christen the 24 other residences spread over 4.5km of the island's northern point.
Some guests prefer the outlook from the Plantation and Seagrass villas right at the water's edge, where the Dr Seuss shapes of pandanus trees frame fantasy island scenes. Or the extravagance of the Peninsula, a three-level villa hewn into a cliff face where the views are best appreciated from a crescent-shaped pool hugging the circular lounge and a plunge pool on the terrace outside the bedroom eyrie. Or the sole (but spectacular) Overwater bungalow, whose sea-facing walls fold away to transform the two-bedroom villa into a breezy platform open to the elements. You can sleep, shower, eat and dream to the soft lapping of waves beneath, and sink into a swimming pool set into a giant volcanic boulder.
But no, I'm perfectly, deliriously happy with the aspect here at Yanuyanu. Kumar politely interrupts my swoon to show me inside the "residence", which occupies three linked bures whose sago-palm thatch droops over their eaves like a Beatles fringe.
After the reconnaissance I saddle up the golf buggy parked outside and zip down the hill to a building of thatched sails that looks like a primitive Sydney Opera House. It is surrounded by a water-world of pools of various heights and shapes cascading into a vast lagoon that meanders past gardens and tiny man-made beaches - just big enough for two - to meet the sea.
Maja Kilgore is waiting beneath the sails of the pool bar to welcome me to this extraordinary island. Few have heard of Laucala because it has always been ultra-exclusive. Owned by the American publishing magnate Malcolm Forbes, it was bought in 2003 by an Austrian called Dietrich Mateschitz, better known as Mr Red Bull. Mateschitz's genius in turning a Thai herbal drink into the tipple of choice for global youth has made him a very wealthy man - the world's 208th richest, according to the annual wealth index ironically pioneered by the island's former owner. The Forbes list pins Mateschitz's fortune at around $4.1 billion, a decent portion of that booty poured, since 2007, into realising this resort of 25 villas and six unique restaurants and bars.
Since Laucala opened in 2009, the world's elite have alighted here on one of the most exclusive islands on the planet. So exclusive that when I visited there were only seven guests in total, with 350 staff to serve them.
There have, undoubtedly, been many celebrities among Laucala's guests, but you will never prise their names from the mouths of staff, whose contracts prohibit them disclosing guest details. Discretion and privacy are the cardinal rules of the island, where an air-exclusion zone ensures paparazzi never get close enough to train their telephoto lenses on frolicking Hollywood couples.
"We have a lot of guests who live in the limelight and they want to be here by themselves and run around in a bikini with nobody photographing them," says Kilgore, who, with her husband Thomas, manages the resort for Mateschitz. Of course, Laucala is open to the public - but only that fraction of the public whose travel budgets run to a minimum $3900 a night. In other words, multimillionaires and billionaires (present company excepted).
The Kilgores ran Relais & Châteaux properties in Germany and Bali before moving to Laucala and spending the next two years realising Mateschitz's dream of creating the ultimate island escape. But it is their dream too. The resort is unique not only for its opulence but also for its self-sufficiency, a philosophy of their making.
More than 80 per cent of the food served on the island is produced there, from the meats and vegetables to herbs and honeys. Even the drinking water, which is filtered through volcanic rock, is drawn from subterranean wells. Basics such as dairy foods and drygoods and premium products like Champagne and wagyu beef have to be imported, but otherwise Laucala's working farm provides for the resort's needs. There are pork and poultry, beef from Hereford-Limousin cows, and lamb from the remarkable "Fiji Fantastic" breed of sheep that shed their own wool and so don't need to be shorn. (There's not a great demand for wool in the tropics, by sheep or by humans.) Natural and hydroponic gardens grow everything from pineapple and eggplant to chillies and the 3500 orchids that adorn the resort. When the Kilgores first arrived they discovered avocadoes, passionfruit, mangoes and even pumpkin already growing there, but they have since expanded that inventory to a kaleidoscopic array of produce.
The surrounding waters - protected fishing grounds - surrender their catch each day to islanders from whom Laucala buys fish, lobsters, octopus and squid. Fresh crabs are easily come by: they clatter ashore each evening and play chicken on the coast road.
As Kilgore is describing the island's bounty, pool-bar staff deliver a parade of tapas-style small bites. A shooter of gazpacho with fresh crab; tuna tataki with mirin and soy; the ceviche-like lobster kokoda; and a lamb cannelloni with tomato ragoût and the most intensely perfumed oregano. Only the flour for the cannelloni had to be brought in. Everything else is endemic, transformed from its raw state into fine dining by Italian head chef Michele Mingozzi, former alumnus of Michelin-starred chefs Heinz Beck and Anton Mosimann and poached by Laucala from the glamorous Raffles Dubai.
"I have never had so much choice before," Mingozzi says of his Pacific posting. "We have five restaurants with five different kinds of food, very separate and very unique. For any chef who wants to cook and be creative and to develop, this is the best place to be." (One day as I am buggying up the hill to a spa appointment, I see Mingozzi fossicking in a glade and stop to say bula. He has a container full of just-harvested fat mushrooms and the vibrant orange blooms of baby zucchini flowers. To an Italian from Ferrara, this must feel like foraging in the Garden of Eden.)
Throughout my stay the food is almost unfailingly impressive, from the off-menu salad of giant octopus, squid and chilli made to order for me at the beach bar, to the best-ever banana ice-cream served at the end of a theatrical lunch at the teppanyaki bar cantilevered over Seagrass Bay. Even the pastries and breads and coffee are excellent. I can't imagine any guests could be disappointed by the calibre of Laucala's cuisine, and that's not something you ever hear said about dining in Fiji.
Laucala's quest for excellence on every level is greatly assisted by the fact that money is no object for Dietrich Mateschitz. Hence he could afford to lure the Kilgores away from their exclusive Relais & Châteaux property in Germany, Mingozzi from Raffles and assistant general manager Simon Hazelman from Namale, another of Fiji's rockstar resorts, owned by the toothy self-help guru Anthony Robbins. Hazelman, a descendant of Europeans who settled in Fiji in the 18th century, is familiar with most of the archipelago's mega-expensive resorts and says, perhaps predictably, that Laucala is "way above them. I came here for a weekend from Namale and was blown away."
Kiwi golf pro Tony Christie had a similar reaction when he was first coaxed here from his life on the PGA circuit to oversee Laucala's 18-hole golf course designed by David McLay Kidd (who, according to those who know, is one of the leading course architects in the world). "I was kind of dumbfounded," Christie says. "Here's a little island tucked away in the middle of nowhere and I land here, and it's 'Wow!' around every corner."
Ordinary mortals like me can't help but have the same reaction. It's not just the fact that Mateschitz has been able to outfit his island with the best of everything, from a flotilla of top-of-the-range pleasure craft moored in the marina to an activity centre crammed with brand new windsurfers, kayaks, dive gear and wetsuits. It is also the attention to detail everywhere.
A full-time staff of 30 gardeners ensures the grounds are never less than pristine. Another seven staff have sole responsibility for the resort's 32 pools. Every restaurant and bar is set up in its entirety every day in case someone drops by, even if there is only one couple in residence. And staff vehicles must always pull off the road (wisely perhaps) whenever they see a guest buggy approaching.
Back at my villa, the interiors are equally pristine. Soft curves replace the usual straight lines and furnishings are - to use another Beatles simile - like something from an octopus's garden. A chandelier's amorphous cream bustle and dangling nacre discs are unmistakably jellyfish-like; fanciful tufted rugs channel sea anemones and the fronds of a floor lamp imitate waving kelp. The effect could be gaudy but the individual pieces are so whimsical and beautifully crafted they inspire wonder rather than ill will.
The décor might broadly be described as primitive luxe, combining traditional Fijian techniques such as roof beams bound with patterned magimagi coir rope, with textured walls of coral-sand plaster, mahogany floors, vanity basins whittled from raintree and bathtubs hewn from rock, without skimping on 21st-century essentials like WiFi, MP3 sound systems, plasma screens and personal mobile phones for guests. (International calls are blocked - wisely - or I would have spent my entire stay on the phone to friends, squealing.)
In the living room bure a bar is equipped with everything from decanters of Scotch and Cognac to bottles of gin, Cattier Champagne and a Vintec fridge stocked with wine. Yet another fridge in the passageway beside the bedroom contains a "health bar" of freshly squeezed juices and iced teas that seems a bit superfluous when there is so much wine at my disposal.
I don't think I have ever experienced so many firsts on one trip before, but that's hardly surprising. Laucala was my first transfer by private aircraft, my first time on a jetski (circumnavigating the entire island - incredibly exhilarating), the first time three rescue boats have been sent for me when I tried to replicate the circumnavigation by kayak (I was fine, they were just being protective) and the first time I have ever experienced what it feels like to be a billionaire, staying on an exclusive idyll I could never afford to pay for myself.
I think it is also the first time I've sat on the edge of an infinity pool, mesmerised by the sight before me, and become misty-eyed at the thought I will never, ever see this place again. But the fact I even got to see Laucala once, and will probably never forget it, is some consolation. There's a Fijian folk song whose words sum up the sentiment: "Beautiful hours are for eternity."