it almost happened during the ice storm this February. It was only a bit below freezing but somehow there was so much freezing rain and maybe some slush where there was running water that the ice literally encased my drive train. About an inch thick in places. I made the mistake of shifting gears and then the chain was skipping over the frozen up cogs and in seconds, it seemed, the chain links also filled up with ice. At that point the bike was immobile.
Fortunately I had a knife and a small screwdriver and was able to chip out enough to free up a cog and the chain. There would have been no calling for pickup - as it turned out it would have been an 8-12 hour drive at that point. I made it home in under an hour though.
One time that I did call, I had no one to blame but myself. I'd ridden a tire a few miles too far and it wore completely through, split with about an inch of tube showing under a few threads. I was only a couple of miles out so I called my wife to bring my spare tire, changed it out, but there was some problem with the CO2 pump I had then. So we wound up driving in.
If we're including accidents, mine involved an aggressive tailgater, a three inch deep rut in the road by the curb, and consequently a flat that I couldn't fix with a broken collarbone.