I keep a bottle of chain lube in my handlebar bag, when my chain gets noisy it needs a but of lube to get comfortable again. That is not really a mechanical, but mentioned it any way.
I built up my touring bikes from parts. I worked as a bike mechanic before I went to college. So, I try to keep my bikes well maintained. And I do not use the really fast tires, get ones that have some puncture protection. So, for me, a mechanical on a tour is quite rare. Punctures, as mentioned in previous posts above is of course an issue. But I have only rarely toured in thorn country, so punctures are also rare for me. I average one puncture a year.
I can't remember any cables breaking on any of my tours, but I have had two bar end shifter cables break at the shifter in the past two decades, I think hand sweat caused corrosion. I now only use stainless cables, no more galvanized.
Lost a cleat screw, not on a tour, but it could have been. I carry spare cleat screws now on a tour.
My Lynskey has replaceable dropouts in the rear. Lynskey did this so that a frame could be built for conventional quick release hubs or for through axle. And two M4 screws hold each of the dropouts in. The drive side dropout is also the derailleur hanger. These screws were installed at the factory and never intended to be removed, but Lynskey did not use locktite on them. One of those screws started to unscrew on my first tour on that bike. And the screw head started to rub on the cassette lock ring. At first I started to hear a funny squeak, but could not figure out what it was. Eventually, that gripped the lockring well enough that it started to unscrew. As soon as the lockring started to unscrew, there was a lot of drag at the rear wheel as the width of the hub was expanding, so I immediately stopped. After a few minutes, figured out what the problem was. Half of the screwhead had worn off, an allen wrench would not fit in it. But this was on a tour, not a ride near home and on a tour I carry a spare derailleur hanger and also a lockring tool. Got a spare screw from the spare hanger. And tightened up the lockring with the lockring tool. When I got home, removed all four screws from both sides, loctited the screws into place.
Somehow, I have no idea how, something got caught in a rear spoke and jammed the against the frame, I felt a sudden jerk in the rear wheel and then continued to roll freely. This was in the middle of Iceland, there were no trees, it was not a stick, the only thing it could have been was a rock. Later that day, at one spot the rim started to rub on the brake. Checked it later at the campsite, and I had a nasty kink in the spoke and the nipple was quite loose. I managed to tighten it up. I had spare spokes and nipples but I did not want to pull the tire and rim tape off, so kept riding. Replaced spoke at end of tour at home. You can see one spoke has a nasty kink in the middle in the photo. The bike is upside down in the photo.
I have to remove my crank arms to pack my S&S bike for travel. I bought self extracting crank arm bolts for that bike. And one of them self-extracted somewhere in the middle of Iceland. So, at end of tour, trying to pack up my bike I had to move the self extractor from one side of the bike to the other side, and of course I did not have the right tools for that. Lesson learned, I now carry a regular shop type crank puller so I can make sure I can pack the bike at the end of tour.
That said, I have seen lots of mechanicals on other bikes owned by others while on a tour. Listing what I can remember below:
My last tour, my touring partner had a flat. A seam in a tube split. That can't be patched, so he rode for a while on his one and only spare before he bought another spare. I always carry two spares on a tour because a few times I have had a non-repairable tube problem.
Rack bolt falling out. Three times on different tours. One of the people that had this problem, I met him in the middle of Iceland where parts as common as M5 screws were unavailable anywhere within a 100 km, he had wrapped duct tape around the rack bolts on his bike and his wife's bike to prevent any loose bolts from completely falling out because he had run out of spares. I now use blue (removable) thread locker on all rack bolts, kickstand bolts, shoe cleat bolts. Other bolts get grease instead. I now even carry a tiny bottle of thread locker on tours. On my last tour, someone at the campground actually asked the two of use on that tour if we had any locktite, he was surprised that I did. Considering that many bike shops do not even have any thread locker anywhere in the shop, bolts coming out is more common than it should be.
In the middle of nowhere, met a Brit that had broken his chain so often that he no longer had all of his gears. I had two spare eight speed quick links, so I could part with one. His drive train was nine speed, I told him it probably would not work, but if he was stranded on the side of the road, maybe an eight speed quick link on a nine speed drive train would get him rolling again?
Front derailleur, either a spring broke or a spring stop broke. I managed to make this functional, just barely with some bunge cord. It took the bike owner a day to figure out that bunge is not as good as a spring, but it would shift eventually if he did not pedal too hard while shifting.
Fairly new bike, external bottom bracket bearing failure. Rider got a new bottom bracket shipped to a bike shop on the route from the bike manufacturer.
Rear derailleur hangers out of alignment, this is most often on bikes that were shipped by airline. Poor indexing and shifting. Several occurrences. I refuse to try to straighten a replaceable dropout when I am in a campground somewhere, as if I broke it, that would be a catastrophic disaster.
A friend of mine went over the handlebars when a steerer tube broke at the fork crown and his lower fork was no longer connected to the bike. The bike had crossed the country on loaded tours twice, and had been crashed a few times. Apparently it had been cracked for a while. This was on a van supported tour, the van driver took him to the hospital.
I know two people that had a rear wheel lock up on them. One on a van supported tour. I was not there so I only heard about it, not sure what happened but it was a cartridge bearing wheel that would no longer turn, he bought a used wheel at the bike shop. The other was not on a tour, her bike rear wheel locked up, it was on PBP and she crashed. She said it was the drive train and they got the wheel to work again. She finished but did not make the time cutoff.
I would not call this a mechanical failure but poor brakes or wheels out of true. In one of these occasions, rubbing alcohol wipe on the rotor fixed the poor braking.
If I think of any more, I will add them. Even though I have never experienced a spoke break on a tour, I will continue to carry spare spokes for the wheels that I built. My light touring bike, I did not build the front wheel so do not have spare spokes and do not know the length, so will carry a Fiber Fix spoke on that bike on tour.