There are no regulatory limits on the altitude that a private pilot can fly. Class A airspace requires an Instrument Rating, but that’s just an addition to your PPL, not a new license.
There are many turbocharged piston airplanes that can get into the low or mid-20s, for not much more than a normally aspirated piston. Turboprops cost quite a bit more, especially when it comes to fuel, but they’ll get you to the high 20s or low 30s.
Above that, you’re looking at jets, and that means a type rating—and a lot more money. Contrary to popular belief, you can get a type rating as a PPL. However, the checkride for any type rating is to ATP standards, so you might as well get at least your CPL along the way, and your ATPL as well if you have the hours to qualify for one. Regardless, that’ll get you to the high 40s or low 50s, depending on the jet you buy.
Beyond that, you’ll have to wait for the next generation of supersonic planes that always seem to be 5 years away. While you’re waiting, be sure to win the lottery so you can afford one—and then several more lotteries to pay the fuel bills.