You absolutely can own them, many people do. But if you fly between 18,000 feet and 60,000 feet, you must fly under IFR rules, and be in contact with air traffic control.

Above 60,000 feet is uncontrolled airspace, however, so you're free to do what you want if you can reach those altitudes. Good luck getting an aircraft that will fly that high, however.

One of the few aircraft I know of that can operate that high is the Lockheed F-104 Starfighter, and there are 15 privately owned ones in the FAA registry. Most seem to be owned by a couple of corporations, but a handful look like they are either owned individually, or perhaps through an LLC (common for very expensive aircraft).