Simple: Go down to Amundsen-Scott Station and walk around the pole.
But if you don't want to cheat, get a very fast airplane:
Jeremy Newton is an Air Force veteran who flew F-18s, but, when contacted by e-mail, suggested the F-22 for a variety of reasons. First, it can fly at 1.5 Mach (about 1,000 mph) without using its afterburner, meaning it burns much less fuel. It tops out at 2 Mach, though that burns more fuel. Second, it can refuel in 10 minutes -- in mid-air while traveling at 400 mph. And third, as the video shows, it can go from full speed to full stop in under four minutes, and to top speed at 30,000 feet in under 5 minutes.
If you don't have access to military hardware, you can still probably hit the four time zones in the United States in that Gulfstream, although you'd be touching down in less exciting locales. (Unless you love the Upper Plains, in which case: go for it.)
The maximum number of time zones you can hit by plane depends on the plane, of course, and on how much you're willing to push it. It seems as though the Gulfstream could get you from GMT+11 to Greenwich Mean Time -- on one tank of gas. The Raptor can do a little better, from GMT+12 to GMT-1, as on the map below.
This is, of course, silly. But it's close to a plan I have on my bucket list: on the June solstice, see the sun rise over Passamaquoddy Bay near Lubec, Maine, and see it set over the Pacific Ocean near the Makah Indian Reservation in Washington. I'd bet you can even do that taking commercial flights.