After 40 years of cycling:
1) all the ball bearings leaked out of one freewheel, fortunately some kind soul gave me a ride.
2) countless broken spokes, but none after I started riding hand build wheels from Bruce Gordon in Petaluma or Vecchio's Bicicletteria in Boulder.
3) a couple of broke chains. no big deal if you have a chain tool.
4) lost rear rack bolt, best to carry spares and use loctite. Weld failure on a blackburn rear rack, a u clamp from a hardware store got me going.
5) a broken seat in England, but they have pubs to help with the discomfort.
6) a broken rear axle in Crested Butte. Don't loosen that quick release until you are at the shop and you probably will be able to ride there.
7) total failure of a Maillard Helicomatic hub in France, fortunately within hitchhiking distance of what was probably the only bike shop that still had a dusty replacement in back.
a broken toe clip back in the day, not a common failure mode in these times.
9) lost a screw on a cleat, lost a screw in a clipless pedal. In Switzerland they broke apart a cleat set and sold me one cleat screw, in Lander Gannett Peak Sports donated a screw from a retired pedal. Thanks.
10) total failure of a front brake cable stop descending Col de la Bonette on our day off, i.e. we left the panniers at camp. Miraculously I was going slowly or else the list could have ended here. TEST YOUR BRAKES, pull on then occasionally very hard while stopped to make sure nothing slips or vaporizes like that cable stop. Look for frayed cables, including inside the brake lever.
11) a occasionally slipping, but not totally clutchless, Shimano LX freehub, it took from Colorado until nearly Nevada to find a replacement. Kudos to Cedar Cycle in Cedar City UT who replaced it after hours, it was a relief to get that fixed before heading across Nevada.
12) a cracked frame racing a thunderstorm and some dirt bikes down Pearl pass. I won, but it was the last trip for the Bridgestone MB1.
13) a broken seat rail leaving Vancouver, Canada. This hurt, but eventually I got to a bike store in Bellingham WA. Thanks to some Sunday truck stop mechanics at the border who donated a hose clamp the last half of that was ridden with a sort of level seat.
14) a couple of wheels trashed by the airlines, in one case the boxed bike was thrown on top of a luggage cart and slid off somewhere on the runway in Denver. The exploded box was described by an airline employee as "opened for inspection by TSA" HAH!
None of these was a calamity, but that cable stop failure could have been the end.