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Næss was born in Germany in 1937 to German physician August Oskar German Raab (1901–1993) and Kirsten "Kiki" Dekke Næss (1907–2001), a Norwegian national whose brother was philosopher and mountaineer Arne Næss. Næss' family lived in Germany during World War II. His parents divorced after the war, and Næss moved to Norway with his mother where they both took her maiden name Næss.
In 1964, Næss moved to New York to work for his uncle Erling Dekke Næss, a shipping magnate and businessman. He started his own business in London in 1968 where he was successful in shipping and oil, and in real estate. He would go on to make a few bad investments in the 1990s in Pyramid scheme Global Money Games. His biggest success turned out to be an investment in an old IT business, Tandberg Data, and various real estate investments around the world. He had built up his shipping empire to be reputedly worth some 600 million by the year 2000. He spent his last years in Switzerland.
In 1966 he married Filippa Kumlin d'Orey of Sweden and together they had son Christoffer and two daughters, Katinka and folk/pop singer Leona Naess. After they divorced, he had a relationship with Norwegian actress Mari Maurstad. In 1985, Næss met American singer Diana Ross on a trip to the Bahamas. He and Ross married in 1985 and had two sons, Ross Arne Næss (born October 7, 1987) and Evan Olav Næss (born August 26, 1988). Through his marriage, he gained three stepdaughters, Rhonda Ross Kendrick (Ross's daughter with Motown founder Berry Gordy), Chudney Ross, and Tracee Ellis Ross; the latter two being Ross's daughters from her first marriage to music executive Robert Ellis Silberstein. The couple divorced in 1999. Næss spent his remaining five years with Norwegian-born Camilla Astrup; they had two sons, Nicklas and Louis.
By age 19, Næss had already made twenty first ascents of Norwegian mountains. He then concentrated on a career in the shipping industry, starting out with his uncle Erling Dekke Næss in 1964 in New York. In the seventies he returned to mountain climbing; in 1985 he led the first Norwegian expion to Mount Everest. Also participating was British mountaineer Chris Bonington.
When asked, during a late-1990s television interview, about the risks of mountaineering, Næss replied: "If I hadn't liked risks, I would rather have played tennis or golf."
Næss was staying with his friend Johann Rupert on 13 January 2004 when he died in a climbing accident while descending from a peak on Rupert's estate in the Groot Drakenstein mountains near the town of Franschhoek outside Cape Town, South Africa. According to police reports, Næss' anchoring equipment loosened from the porous mountainside, leading to a 103m fall. He was 66 years old.