What is the most serious thing that has happened to your bicycle while you've been out on a ride?
I've had lots and lots of flats, a torn tire, shifter cables that seized up on me, and broken spokes. My chain has fallen off several times, as have my waterbottle cage, and lights. My handebars loosened on me one day and dropped forward. I've had a crank arm fall off, and a pedal fall apart. And my seatpost cracked (although I discovered it later)
But the most serious was when my freehub seized up while I was on a tour. When that happens pedalling does not engage the rear wheel anymore. Fortunately I was not far from where I was going when that happened, so I coasted in, but I had to replace the freehub the next day.
How about you ... what is the most serious bicycle repair you've had to do while on the road?