I've seen people use rolled foam like this rolled around the inner edge of their frame-less backpacks for a little support. They take up very little room this way, because it's just the thickness of the foam x however many wraps it makes around the inside of the pack. The "inside" space is used storing the rest of your gear, the added support is nice on your back, and it also protects the thinner fabric of the pack from anything sharp inside.

It's definitely important to have something for a sleeping pad, although imo the inflatable ones work better. If car camping another useful thing is to bring loose cardboard you have lying around and line the ground with it before setting up the tent, adds an extra level of comfort on top of the sleeping pad.

Here's maybe a weird use for this material: If you get a completely smooth foam mat (instead of textured as seen in the photo) you can use spray glue and glue down a sheet of sandpaper. (Or multiple sheets of different grit levels)

Then, on the sandpaper, you can sharpen your knife by hand and give it a strong convex edge instead of a v-edge that you would get from a stone or plate. The "give" of the foam allows the convex shape to be formed in the steel.

A purpose-built seat pad is not that expensive, and it's already an appropriate shape - no cutting required