I carry some to most of the stuff listed on this thread so far (not all the time or at the same time) depending on the area I'm going to, or length of trip. But, I do carry one thing that I hadn't seen listed on this thread.

I carry a Walking Cane on my bicycle. Whether it be a full size cane, or a Folding Cane. I can ride a bicycle much better than I can walk, so if my bike suffers catastrophic failure (for whatever reason), and I find myself on foot, at least I can hobble easier with a walking cane.