I've been a private pilot for many years, but have never flown internationally in small private aircraft. I'm about to buy a tiny, high-tech, super-efficient 2-seat airplane that has extended range fuel tanks that make it possible to fly 8000km == 5000 miles without refueling. This makes it possible to fly anywhere in the world, though stops are required to cross the south pacific. Besides, you'd be nuts not to stop after sitting still so long (about 15 hours per 4000km/2500mi). In fact, for most flights I'd prefer to stop every 2000km or so for a meal and/or potty stop and/or stretch the body and legs.
This raises the asked question. Can a pilot, with or without one passenger, stop in other nations just to refuel without needing to go through immigration (and have a visa for that nation) just to land, pull up to the gasoline pumps, fill up, get back into the airplane, and fly away?
Note that this is NOT a question about flying in a commercial airplane on a conventional scheduled airplane flight. I also presume the stop will not involve going inside any building or leaving the refueling area within the private aircraft portion of the airport. It will involve getting out of the airplane, because refueling of diverse small private airplanes is generally done by hand by pilots, not by the huge trucks that fuel jet airplanes. Typically small airplanes just pull up to a pump much like you'd see at any gasoline station, insert credit card, pump fuel into the wings, then fly away.
When I get the airplane, after a month or two of getting used to it in north america, I plan to flying down to Chile with 1, 2, 3, 4, 5 stops between (depending on these rules), then fly to dozens of islands in the south pacific (and land many places and islands that have runways, but no buildings (or no legal/government buildings).
In the later cases (south-pacific islands with tiny populations), I suspect nobody bothers or cares, because they assume all air traffic is strictly local. But flying down through (or 20 miles offshore) of Mexico through central America then Columbia, Ecuador, Peru and finally northern Chile (where I intend to spend a great deal of time) may be another matter entirely.