Bouvet Island or Bouvetøya is one of the Subantarctic Islands, an uninhabited ice-clad place 2,600 km south-southwest of Cape Town. It has a good claim to be the world's most remote island: anywhere within 2,000 km is similarly desolate, such as mainland Antarctica. It lies 6,040 km south of the equator and is a dependency of Norway; for comparison Oslo main railway station is 6650 km north of the equator and can be cold but never ice-clad. That's testament to the chill of the Southern Ocean surrounding Antarctica, compared to the balmy North Atlantic.
Bouvet stands on the mid-Atlantic ridge, where the continents are moving apart, so it formed as a volcanic island in a similar way to Tristan da Cunha and Surtsey. It extends 9.5 km east-west, 7 km north-south, with its highest point Olavtoppen or Mt Olav at 780 m, and a deep caldera just south of the peak. The last eruption was probably around 2000 BC. The rocks only 30 cm below ground are at 25°C, but the island is almost completely ice-clad, and the volcano appears to be decaying into extinction.
The island was first sighted on 1 Jan 1739 by Jean Baptiste Charles Bouvet de Lozier, hence its name, but he recorded incorrect coordinates (setting a trend) so the place was lost. It was re-discovered in the 19th century, landings were made, and sealing ships came to the area. But with no sheltered harbour and with daunting 500 m cliffs around it, Bouvet Island could never become a whaling base like Grytviken on South Georgia. The first extended stay was by the Norvegia expedition of 1927; Norway made further expeditions and in 1928 claimed the territory. Britain protested that they'd planted the Union flag in 1825, but the co-ordinates and logs of that voyage were garbled. (And a US "landing" of 1823 was obviously bogus.) The place wasn't worth spilling diplomatic ink over so Britain abandoned its claim and Bouvet became Bouvetøya. Norway has two other dependencies but both lie south of 60°S, where such claims are suspended by the Antarctic Treaty. So insofar as Norway boasts an Empire, this is it right here.
In the late 1950s a landslide created the area of Nyrøysa, which translates as "new pile of rocks". It's a coastal scree that is more accessible and free of ice, so a research base was established in 1996. An earthquake and storms in 2006 erased it but a new base was installed in 2014. The island and its adjacent territorial waters have been designated as a nature reserve since 1971. It lies within the Southern Ocean, south of the "Convergence" with a climate that's cold but without polar extremes: it averages around 1°C in summer and 3°C in winter. There's a huge colony of macaroni penguins, and there are other penguins, southern fulmars, and seals and whales in the waters around.
The island is 2,600 km south-southwest of Cape Town in South Africa, far from cruise ship routes. It's walled by cliffs, with nothing resembling a harbour, though an expedition ship could anchor to leeward. Then you could approach in a Zodiac or similar small craft and attempt to scale the cliffs. Likewise there is no landing strip, but it's possible for ship-based helicopters to set down, though there's often fog and wind. Those who've visited and lived to tell the tale have mostly got in and out this way. Frankly it might be easier to join the Royal Norwegian Navy Marinejegerkommandoen, distinguish yourself in training, and persuade them to land and camp on the island as a team-building exercise.
You may be able to join an expedition if you have special polar skills. One surprising example was DXpedition, when amateur radio enthusiasts set up a radio station to communicate with people around the world, and future visits are planned. Read up on current topics and projects, and also the 2016 / 17 polar policy paper adopted by the Norwegian parliament. For example, they might be interested in micro-plastics and similar pollutants in a place so far from anywhere.
In theory you need permission to visit from the Norwegian Polar Institute. In practice they don't ask for this because they know it's next-to-impossible, and any expedition must involve other activities (such as helicopters) that do require permits.
Almost 93% of Bouvet is covered by glacier, riven by crevasses that are not charted. So you need the skills and technique to cross such terrain and it might be simpler to call on the helicopter that brought you in to shift you. The small ice-free area at Nyrøysa is a special area of study—you may not enter it Nov–March except in an approved monitoring party.
Anticlockwise: