I keep a fanny pack around (for when I don't have enough pockets and to keep stuff in one place until I stuff my pockets ) with extra money, some ID, a cell phone, a couple of food bars, some Endurolyte tablets, a pair of rubber surgical gloves (for keeping your hands clean when fixing flats, putting chains back on etc.). Some guys keep a wad of duct tape, a few zip ties, and few other odd bits for on the road repair. You've most likely got Schrader valves on your tires but if you ever decide to go the road bike route a couple of those Presta/Schrader adapters can come in handy when you've used up your last inflator kit cartridge and nobody has a Presta valve pump anywhere in sight.
If you start using clipless pedals, then a couple of popsicle sticks to clean the dirt out of your shoe cleats are handy.
Another handy thing that I've learned to do while xc-skiing is to put anything that can get wet into a zip lock bag. Stuffing the extra little bits into one that you can stuff into a pocket is a good idea too. They keep stuff in one place, they prevent the pointy bits from putting holes in your pockets and everything stays dry. I've got hayfever so I never go without a couple of those little travel packs of tissues. The plastic packs keep them dry and they don't take up much space. Some people carry baby wipes in a zip lock.
See
http://www.bikeradar.com/gear/articl...repairs-18199/ for some interesting hints.
As for what breaks aside from getting flats ? Derailleur cables. Check them as part of your checkout ritual before you go out. When I was a teenager, a friend of mine had a cable snap while a bunch of us where doing a long climb in the middle of nowhere. We got him back on the road and had things working well enough but it would have taken a lot less time if I knew what I know now.
Anything else ? I always ride with gloves, usually the fingerless type but if it's cold I'll stuff a pair of full fingered gloves into a jacket pocket. One thing I should be carrying is an Epi-pen especially when I'm riding in unpopulated areas.