I generally just carry a knife or two, or perhaps a knife and leatherman, but then I'm usually using a stove and not relying on fires, except in extremis or for ambiance. If weight isn't such an issue (as in sea kayaking), and I expect to be using much wood, I might carry a small saw, and possibly a small hatchet. The knife/saw/hatchet combo is very versatile.

I'm partial to the Sven Saw. At slighly over a pound, it's a bit heavier than the Bahco, but has a 21 inch blade. One can do some serious cutting with the sven.

Unless car camping, where weight and bulk is really not an issue, I wouldn't carry a larger axe. Unless I were planning on building a cabin, I can't think of much that I would want to do that couldn't be readily done with the knife/saw/hatchet combo. But then if I'm car camping, I could just as easily throw in my little chain saw!