I'm a big fan of traveling to some extreme points, like for example the most northern point of Europe or the highest point of each country in Europe, etc. To find new ideas I would like to know the most remote exclave, island, etc of a Europe country, and also worldwide.
You could travel to extremal points of the European Union. The southernmost point of the European Union is on the Île de la Réunion, in Saint-Joseph. The easternmost point is on the same island, in Sainte-Rose. The westernmost point is in Saint-Martin (the French part, since the Dutch part is not in the EU). If you're looking for the southernmost or westernmost continental points, they're in French Guiana. If you want to spend your euros, you can go further south and east to the Kerguelen Islands.
If you want to leave the EU while remaining in an EU member state, you can visit the UK, the Netherlands and France's other overseas territories, as well as Greenland (which contains the world's northernmost point on land). Greenland used to have the northernmost point of the EEC, the EU's precursor. The world's southernmost point with an undisputed claim (i.e. not counting Antarctica) is in the South Sandwich Islands, a dependency of the UK.
If you're looking for “strange” exclaves, you could visit exclaves inside exclaves: Baarle-Nassau close to home, and a few others throughout the world, including an exclave in an exclave in an exclave in Cooch Behar on the complex border between India and Bangladesh.
If you like mountains, you might want to visit Lesotho's lowest point: at around 1400m, Lesotho is the country which is furthest from the ocean level, far above the next contenders, Rwanda and Andorra. And conversely, you should visit the hills near the Dead Sea, which don't all quite rise above sea level.
Tristan da Cunha archipelago is the most remote inhabited place in the world.
I see your Tristan da Cunha, @mouviciel, and raise you with the aptly named Inaccessible Island. Although not strictly inhabited, people have been there before. Just about impossible except by fishing or research vessel out of Cape Town.
Great post by Giles. I can only add Saint Pierre and Miquelon in North America
"Ushuaia" es la Ciudad Más Austral del Mundo. No se discute.
"Puerto Williams" en Chile es una aldea casi una base de casi 2.000 habitantes. Ushuaia una verdadera Ciudad de 57.000 habitantes. Y si los ingleses nos devolvieran nuestras Islas Sandwich del Sur (hoy dependientes de Gran Bretaña a la fuerza ilegítima), cuya isla Thula más al Sur está a los 59 grados de Latitud Sur casi tocando el paralelo 60, ENTONCES la República ARGENTINA ES EL PAÍS MÁS AUSTRAL DEL MUNDO.
Chile el segundo más Asutral con la Isla Diego Ramirez; pero la Argentina el país Más al Sur con el Destacamento humano "Corbeta Uruaguay", destruído por los ingleses en 1982 (ese sería el asentamiento humano (y desde ya argentino) más al sur del planeta sin contar a la Antártida).
"Ushuaia" is the southernmost city in the world. This is not disputed.
"Port Williams" in Chile is a village of almost 2,000 inhabitants. Ushuaia a genuine city of 57,000 people. And if the English returned our South Sandwich Islands (today dependencies of Great Britain due to illegitimate force), whose southernmost island Thula is at 59 degrees of latitude almost touching the 60th parallel, THEN the ARGENTINE REPUBLIC IS THE SOUTHERNMOST COUNTRY IN THE WORLD.
Chile the second most southern with Diego Ramirez Island; but Argentina the country furthest to the south with the human detachment "Corvette Uruaguay" destroyed by the English in 1982 (that would be the human settlement (and right now Argentine) the most far south on the planet excluding Antarctica)...
(apologies, we've left some words as there by the translator as we didn't want to risk changing the intended meaning, even if it wasn't totally clear to us)