While not on a tour I did have an early generation mtb bull moose handlebar weld fail right at the stem. Usually a very strong union and the bike was used as a commuter. That would have not been field repairable until in a town for welding or replacement. A snapped seat post could also qualify, especially the modern ones that can't be jerry rigged. I suppose you could stand and pedal as long as possible.
With my fat bike a wheel problem could cause a longer than usual delay as most stores don't carry replacement sets of hubs or rims, in out of the way places it might even be hard to get a tire.
I have seen a pedal strip out a crank arm before on my brothers bike. Probably due to cheap metal in the crank and seizing pedal bearings. The female threads were completely gone. No idea how he did it.
Last edited by Happy Feet; 11-12-19 at 09:47 AM.