Andy Hall recently drove his Tesla Model 3 from his home in Big Horn, Wyo., to Washington, D.C., for his daughter’s graduate-school commencement ceremony. He turned what is normally an 1,800-mile affair into a 10,000-mile, 15-day odyssey with more than 80 stops in places such as Miami Beach and Quebec City.

Dr. Hall, a retired ophthalmologist, is part of a cadre of die-hard Tesla owners who are racing to visit as many of the company’s fast-charging stations, called Superchargers, as possible. It’s a competition without a prize or even a finish line.