This is classic "bad battery". It doesn't have the cranking amps to start the engine. Hence the, "rrr-rrr-rrr" sound. Then, after a number of attempts to get the car started the battery ends up completley dying...charging doesn't help because the battery is shot.
I agree, other than not starting, there is no symptom of a bad starter described.[/quote]
Winter usually reveals weak batteries....If it's more than a couple of years old, I agree that might be the best place to start..
However, it its cold starter solenoid can stick and prevent the starter gear from engaging the flywheel. When a solenoid doesn't work properly...it can make a clicking noise just as funboy described. (Older cars had the solenoid as a separate part from the starter...newer cars had the solenoid built in as discussed by nutdotnet.)
Back in the day...I carried a small ball peen hammer to tap the starter solenoid / starter to help it engage...because it had a tendency to stick when it was cold. I don't think that you can get to the starter of a car, easily, now.