I think I would also vote for a sleeping bag (polarguard to boot). An axe would be a bit of a luxury wouldn't it? I've backpacked for weeks at a time and never needed an axe. But a sleeping bag? I would live pretty comfy with one of those. Of course I'm assuming I have a knife and fire starting equipment.
Anyway, getting a good night's sleep will help you keep from going totally mental. Also a sleeping bag is perhaps the most expedient treatment for hypothermia.
I guess there are places where an axe could be essential. If I had to build a long-term shelter, for instance. An axe will build you a nice little trapper cabin. Say your plane crashed in the middle of nowhere. You might use an axe to build a dugout canoe that could get you to safety more quickly (you would expend a lot of energy though). But aren't you supposed to stay put in those kind of situations?
And with an axe, you can get into the interior of old trees for dry wood where, under heavy rainy conditions, dry wood may be tough to find.
But my idea of fully equipped survival would be all the backpacking essentials that I usually take that allow me to live very comfy. If I was going to add a large item to that, it would be some kind of gun, maybe a 22 rifle or a single shot 410. In good fishing country, though, I doubt I would need it. I can live on fish for a long time and supplement with wild edibles.
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Hoodoo
No, I do not weep at the world--I am too busy sharpening my oyster knife.
Zora Neale Hurston
Oversharpen the blade, and the edge will soon blunt.
Lao Tsu