You wouldn't believe but I never carried any sort of anything, not tubes, not a pump, not a water-bottle, nothing, until this past spring. I laid down a lot of fairly trouble-free miles in between age 13 (when I first started riding a lot on the roads) until this past spring, though I did have a couple of walk, walk, walk, walk occasions where I lost a tube someplace far from home. The simple expedient of purchasing a $25 hand pump and an $8 spare tube for the mtb turned a couple of potentially annoying days into nothing remarkable. Just a quick changeout and I was back up and running. Having water bottles is a great improvement, too. I had no idea how much more pleasant it would be and how many more miles I could stand with some water handy. It even makes my little 3.5 mile ride to work easier, especially when it is really hot out.

I really kind of hate the frame pump I have for my road bike. It's a Blackburn, which is supposedly pretty good, but I find it wobbly and hard to hold steady, which means mangled presta valves every time. I'd almost rather walk, honestly. Plus it takes ages and ages and my arm eventually gets tired.

Nowadays, I carry an allen key set, a folding screwdriver with flat and Phillips heads, a tin of band-aids, a patch kit, a spare tube, Presta Valve extender for said tube, a bottle of sunblock, a chapstick, tire levers, and some extra hair ties.