Mount Pleasant in the Falklands I suspect
I think the current RAF Airbridge schedule, open to civilians, is leaving Brize Sunday & Wednesday and returning Mon & Thurs night - round 24 hours door to door each way
The LAN Chile commercial flight leaves Santiago at some ungodly hour every Saturday morning via Punta Arenas and sometimes stops in Rio Gallegos - gets you in around 14:00 and back to Santiago by 22:00
there is also an oil field trash charter out of Gatwick every couple of weeks while they are drilling - which they are now - its the shortest trip as they stop in W Africa rather than Ascension - don't think its open to all comers - might be a Thursday down and Friday back but don't quote me