I carry The Max Ax which is a multi-purpose ax, shovel, pick, rake, etc. Got it for $75 from a CL seller that had several military sets. It's been good for splitting firewood.
One thing I noticed is that no one recommended one of those "wildland firefighter" axes, aka a Pulaski Axe. The one with a digging blade opposite the axe blade. Strikes me as being a useful supplement to a shovel, without having to carry a separate pick-mattock or one of those expensive multi-tools. If you think you need that kind of digging capability . . .