I keep the sticky strings, and the T-handled tools (rasp and rope inserter). The GL1800 tires, especially the rear tire, can be a tough customer.
I also keep a small tube of rubber cement, as it seems to me, it acts as a lubricant when inserting the strings. It might also help to seal the edges of the puncture as well.
A Leatherman multi-tool as well.
For air, I bought a cheap Campbell Hausfield 12 volt compressor, and took it out of the case, reduced it down to its guts. It now easily fits inside of a 1 quart freezer bag, with room to spare. I extended the power cord so it can pump up the front tire, while plugged into a cigarette style outlet in the trunk.
It can inflate a tire from completely flat to 40 pounds, in a few minutes.
Of course, you can always get roadside assistance and a cell phone. My buddy had to go that route, when he picked up a 3/8" bolt in his rear tire, that he couldn't even pull out with a Leatherman multi-tool... and the tire only had 38 miles on it.