Things requiring either roadside action or cutting the ride short.
Not counting crash damage or things discovered after investigation at home, such as a cracked crank or a worn out BB.
Spoke broken
Pedal failures
Frame cracked
fallen/falling out bolts
tyre cuts/excessive wear
chain jammed behind crank
broken chain
broken mudguard
broken gear cable
broken crank
split/worn out rim
bent rim (flat spot from a pothole, bad enough to make the brake unusable)
broken handlebar
I ride 36x36, so a broken spoke generally just requires action to stop it flapping about, though I will fix it at the roadside if I'm suitably equipped (i.e. on tour). IIRC, it's 4 in my first machine built wheel, 2 impact damage, 1 at the nipple with 26"/Rohloff/3x, and about 10 that broke in the middle, that I put down to a bad batch of wire at the manufacturer (plus one possible pending failure - I noticed a bend/wiggle in a driveside rear spoke the other day).
The pedal failures were all Eggbeaters, or the Look 4x4 clone. One snapped axle, and several bearing failures that allowed the pedal to slide off the axle. Shortest route home generally, but I ended up carrying a replacement bearing for a bit, before I gave them the sack.
The frame were 3 x gearside dropout breaking where it plugs into the chainstay (fixed under warranty by SJSC), 1 x seat tube/BB shell join in a very old secondhand Mercian, and one cracked round the down tube, starting at the lug point. All were rideable, in a directly home fashion.