I've been a private pilot for many years, but I 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 (24+ hours). The longest flights necessary to cross the south-pacific in hops is 4000km (about 16 hours of flight at 250kph).
Such long range makes it possible for this tiny aircraft to fly anywhere on earth. Nonetheless, I'd have to be nuts not to stop more often than every 16 hours when that is possible: for a meal, potty stop, walk around and stretch body & legs... and eventually sleep!
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 get a visa for that nation)... just to land, pull up to the gasoline pumps, fill them up, get back into the airplane, then 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 airstrips, but no buildings (or at least 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 then finally into northern Chile (where I intend to spend a great deal of time) may be another matter entirely.