The worst I ever had was a total tire blowout. Not just a normal flat tire (i.e. a hole in the tube), but the sidewall of the tire ripped open. Not at all patchable on the side of the road.

What made it worse was I was going through a sort of park area, and while houses were visible up and off to the side, I had about a 2 mile walk to get out to anywhere useful, and walking that kind of distance in cycling shoes is hard on the shoes. As it happens, this took place in warm weather, but that event spooked me from riding that trail in the winter. For a while I tried to carry an extra pair of shoes (bulky/heavy) and an extra fold-up tire (bulky/heavy).