I was riding to work once and my free-hub seized up, locking the rear wheel, and shaving the tire down to almost nothing. How would you fix that without help, on the road? (I locked it to the nearest safe post, took a cab home, and drove to work.)
Problems that can come up are like bikes; they follow the N+1 rule, with N being the number of problems you're prepared to solve. I swear to god Murphy's law plays into it, too - you won't get as many flats if you have a spare tube, but that will just make the bike gods find other ways to taunt you.
What I'd really like is a shuttle service. I want to do some rides that are too long to take as a round trip, and, believe it or not, most of my friends have better things to do than drive me around. If I could get someone to drop me off with my bike in Winthrop, and meet me at Diablo Lake, or move my car there, it would be the cat's ass.