The little pocket-knife multi-tools sometimes come with tiny saws on them. Those saws are somewhat useful but extremely limited in the size wood they can be used for.
Are there any multi-tools available which include bigger tools, including a saw big enough to be very useful compared to their tiny cousins, which are still small enough to be pocket size or possibly just a little larger?
Some of these pocket-knife-saws are only 2 or 3 inches long. The sticks they are useful for are barely out of hand-breaking range. If we had a multi-tool that doubled that size, say 5 or 6 inches at least, that would be immensely more useful, opening up a much wider range of sticks/branches it was appropriate for, and it could still fit in many peoples pockets or on a belt.
I ended up buying a pocket chainsaw in response to Charlie's answer. The pros and cons are large, so I'm sharing the results with you.
Pro: The pocket chainsaw packs down very small and is not heavy. I decided to keep this as the saw of choice for the kit in question. Con: I bought a decent one, yet the pocket chainsaw does not seem to be nearly as good as a normal saw.
I shopped around, read product descriptions and customer reviews and watched videos of various pocket chainsaws being used. I bought what looked like one of the better ones which multiple people were swearing by, and it even comes with a lifetime guarantee.
I tested this saw on some 4-6 inch trees and branches. I was able to cut them up, which I couldn't have without a saw, but it took way longer and was more tiring than the same work with a normal saw. And it was very jarring; the cutting teeth stick out more and rattled my body with every stroke, leaving me sore. And the cutting motion and handles were awkward and left me with nasty long-lasting blisters on my fingers.
So I would not recommend this pocket chainsaw as being better than a conventional pruning saw, bow saw, or other hand saw by any means, but it is a great addition for the purpose intended in this question. The pocket saw has drastically increased the range of wood I can cut with my tiny outdoor kit.