Sure, super-high-tech Japan might be the island with the most remotes in the world, but the distinction of being the most remote island in the world goes to the South Atlantic's Tristan da Cunha. It's a full seven-day boat ride from civilization, is perched on an active volcano, and has no traditional hotels... but is still (actually) a viable place to visit!
Buried DEEP in the South Atlantic, Tristan's 2088mi from South America, 1750mi from South Africa, and 1500mi from Saint Helena, which's the nearest landmass of any kind.
Although you'll have plenty of time to read up on it during the seven-day boat trip from Cape Town (literally the only way to get there), the island was named after an explorer who first spotted it in 1508. In 1810, it was briefly renamed Island of Refreshment by its first settler (and self-proclaimed "ruler of the islands"), but that expat American died in a boating accident two years later, and the name was refreshed back to Tristan da Cunha.
After seven days on a boat, you'll feel mighty relieved when you first see that volcano on the horizon. But then you'll be even more relieved when you see actual civilization in the island's only flat-spot/town, Edinburgh of the Seven Seas. Population: 276.
In the event you forget the fact that it took you seven days by boat to get to an island that's settled by 80 total families who share seven total surnames (Glass, Green, Hagan, Lavarello, Repetto, Rogers, Swain), this sign is a helpful reminder.
There's just one doctor (and five nurses) on Tristan de Cunha, but the place is technically a British colony, so those credentials are pretty legit. That said, the closest place to treat a serious injury is Cape Town, so if this happens to you, you're really screwed.
Should you remain healthy, you'll need a place to stay. In lieu of hotels, locals rent out guest houses or rooms in their own homes (usually on an all-inclusive basis), for $20-$30/ night.
Once you're unpacked, you can walk through town... in 15mins.
If you aren't exhausted by those 15mins of exercise, stroll through their farm fields, or, uh, climb up into the cliffs? Just remember: nearest hospital = seven days away.
Classic Tristonian ruffians. Don't mess with these guys! Or stay too long, because you'll quickly become one of them.