Tristan da Cunha
Tristan da Cunha is an archipelago of fairly small islands in the middle of the South Atlantic Ocean. It is a territory of the United Kingdom, administered by the government of St Helena, which lies several hundred miles to the north.
Tristan da Cunha is the most remote inhabited island in the world — the nearest speck of land, St. Helena, is a whopping 2430 km away, and it's over 2800 km to the nearest continent, Africa.
The entire population of some 270 inhabitants is concentrated on the only flat bit of this volcanic landmass, the hamlet of Edinburgh of the Seven Seas on the main island. There are a few other islands in the archipelago, all uninhabited: Inaccessible Island, Nightingale Island, Middle Island and Stoltenhoff Island. Gough Island, some 300 km away, hosts a weather and scientific research outpost.
All visitors to Tristan da Cunha must receive a permission from the Tristan Government. Write an email to email@example.com and specify when you plan to go, where you intend to stay and the purpose of your visit.
There is no airstrip on Tristan da Cunha.
Travelling to Tristan da Cunha requires careful planning. It takes five to six days to travel the 2810 kilometres from Cape Town. The South African polar research ship SA Agulhas and the fishing vessels Edinburgh and Baltic Trader do the voyage between Cape Town and Tristan da Cunha several times every year. A return ticket on Agulhas is about US$1300, a return ticket on one of the fishing vessels is US$800. Schedules and further information is available on the official Tristan da Cunha website.
Due to rugged, steep terrain, going all the way around the island is difficult, but if just staying in the village of Tristan, the flat, grassy ground is easy to maintain.
There is a paved road (the M1) from Edinburgh (aka The Settlement) to the Potato Patches, which are about 3 miles away. Local transport is available to the Potato Patches. This local transport could be an islander's car, tractor, and during the mornings a bus service also operates. Note that the bus is targeted at pensioners, who can ride on the bus for free. The charge is £5 return  . Note that you cannot rent any vehicles on the island.
The Island organizes fishing excursions, walks, climbs and even golf for visitors. Once again, consult their website for more information.
Take a trip to Gough Island. Gough Island was first known as Diego Alvarez, but it was sighted again in 1721 by Captain Gough, from his ship the Richmond. This brought a new name - and a bit more attention - to the place. Though Gough Island is a U.K. territory, the only permanent settlement you will find is South African. South Africa leases a portion of the island from the U.K. for use by SANAP  as the only permanently manned South Atlantic Ocean meteorological station . The island is a UNESCO World Heritage Site.
Gough Island has no sheltered harbour or anchorage. The only suitable landing place for boats is at Glen Anchorage in Quest Bay on the island's east coast.
SA Agulhas, on a relief expedition, departs from Cape Town to Tristan da Cunha then onwards to Gough Island on an annual relief voyage. This ship carries cargo and passengers.
There is presently no access for tourists and even crew members from passing yachts may not go ashore except in the case of an extreme emergency.
Getting around comes with great difficulty - combination of excessively steep terrain and incredibly dense vegetation - and no paths to speak of. There are no public accommodations on Gough Island as well.
Eat & Drink
The only public place available is the Prince Philip Hall which occasionally serves food, the building also houses the Albatross Bar - the islands only pub. Opening hours are sketchy to say the least, and the only time it's very likely to be open is when cruise ships are docked at the island. If you are hungry and the hall is closed, your only other bet is a visit to the Island shop.
Self-catering accomodation is £20 per night, while home stays, which include meals and laundry, cost £40 per night. There are discounts for Tristan Islanders and children. Booking information is available on the Island's official website.
Relatively nearby to the south is Gough Island, another British dependency.