The Nürburgring, one of the most famous race tracks in the world is located in Nürburg, Germany. The Nordschleife track is the most popular at Nürburgring, in fact it's what most people think of immidiately when you say Nürburgring.
Nürburgring allows the public to access Nordschleife, open mainly on Sundays, but also many Saturdays and weekday evenings. The public has been able to access the course since 1927. Anyone with a road-legal car or motorcycle, as well as tour buses, motor homes, or cars with trailers are allowed to race on the course.
Of course, allowing anyone on the track comes at a price. Many members of the public that want to ride on the track are inexperienced and end up crashing. The good news is that normal, non-racing, non-timed driving accidents are usually covered by driver's insurance. Driver's insurance is voided when there is an attempt to race.