Two (OK, actually three) submissions here - the first is Swan Island, owned by Honduras. Anyone who has flown into Roatan or flown into Honduras, El Salvador or Nicaragua has probably flown over it - it is "Cisne", as in "Isla de Cisne" on the charts. Nothing but a short (1,500 ft) coral strip and at one time a radar site for tracking the druggie traffic. The island has the highest population of iguanas per square mile than anywhere else on earth. On landing you'd squash at least 3 or 4 of 'em. While we were on the ground we'd go snorkeling - talk about an "undiscovered" coral reef! After seeing our fill of tropical fish we'd load up and head out. It was like flying onto Gilligan's Island.
For actual instrument approach fun, I say the now defunct Hong Kong curve. Nothing like flying an ILS to the top of a building and then turning at the checkerboard to line up on the always-with-a-strong-crosswind runway.
Finally, for absolute "What the f***?", it has to be Grand Cayman, and in about two weeks or so. Sometime in the late spring it becomes crab mating/migration/block party season, and the runway and taxiways are covered in crabs. Literally. On taxi out you can hear and feel them getting crushed under the nosewheel. There are thousands of them on the runway, so there is no way to avoid them. As we barrelled down the fairly short runway in our trusty 727 and approached V1 I wondered if anyone had figured out what the RCR of crab guts on the runway is. From my experience the inside of a crab is pretty slimy, and slimy means an RCR of about 1. I pictured us losing an engine and skidding into the water at the end of the runway, only to be asked later if we had taken the crab guts into account in our takeoff planning.