Well, that was easier than I expected.
Last Thursday I got rear-ended in traffic while driving home from work. Minor damage was done to the back of my Subaru Outback, but quite a bit was done to the front of the little Mitsubishi Mirage. That evening I rang the insurance company, talked for about 15 minutes (after being on hold for about 10). Friday I took the car out to the inspection place, they got a taxi to take me in to the city. I made the police report over the counter in the city and they entered all the data for me. Today (Monday) I got a call from the repairers, car's done, so they sent a taxi for me. Got the car back, looks nice, and apart from the notes I took at the original incident, I haven't had to set pen to paper about anything and haven't signed a darn thing.
Not too bad, really.