Høibakk is a noted mountaineer. In 1958, he was part of the first climbing of Trollryggen. He also climbed the mountain Tirich Mir in an expedition led by the Norwegian philosopher Arne Næss in 1964. In 1985, Høibakk climbed the Mount Everest with a group of 25 Norwegians, led by Arne Næss, Jr.. Ten years later, Høibakk climbed the Drangnag Ri mountain in Tibet with Chris Bonington and Bjørn Myhrer Lund. Høibakk and Bonington were hit by lightning at above sea level. In 1990, Høibakk skied to the South Pole with a group of Norwegians who were the first Norwegians to reach the South Pole on ski since Roald Amundsen. He later recalled that he found the ski trip a bit boring.