In the US, pilot’s are issued Certificates, not Licenses, and very few states require licenses any more. A ‘license’ is issued by a state or local government to ply some trade in that place, and many of them require certification by some authority, like the FAA or the trade associations for barbers, electricians or plumbers. Pilot certificates are issued after you pass the FAA’s written exam and a checkride with the FAA.
A private pilot can fly any plane they can afford to rent or buy. An instrument rating is needed, and if it’s a turbo jet or any airplane that weighs more than 12,500 pounds a ‘type rating’ is required to be pilot in command. Some jets, smaller and simpler, are certified to fly with a single-pilot, others require a co-pilot, and some older or larger jets require a flight engineer.
John Travolta might be the poster boy for ‘largest private jet’ owned and flown by a private pilot, a Boeing 707 he got from Quantas and maintains at his house, on a private airport. When he wants to fly it, he gets a crew together, a co-pilot and a flight engineer are required to fly with him.
He has several jets, and is ‘only’ a private pilot with type certificates to fly his jets. Some of them are single pilot, others require a crew. Notably, he experienced an avionics failure in the air over New York and managed an emergency landing for his Bombardier Challenger at Teterboro and taxiied up to the radio shop with no problems.