Carbonized cellulose acetate has tiny pores... so tiny that they perform better than graphene or carbon nanotubes as a supercapacitor medium.
Where can you get cellulose acetate? You can make it... get a fine-grained wood and grind it to a fine dust. Then "de-lignify" it with a basic solution and heat/pressure.
Or use a brown paper bag, which is made of wood pulp and contains nearly no lignin. The binders in a brown paper bag (if any were added) might throw a wrench in the works. I've not tried it yet, but a gentle rinse in acetone should rinse out the binders. Let it dry before proceeding.
Then apply acetic anhydride. You'll have cellulose diacetate. If you used wood dust, it'll be in powder form. If you used a brown paper bag, it'll be in a sheet.
Then (if you've used wood dust) apply acetone and form the powder into whatever form you like (wafer, etc.).
Then place your wafer or whatever shape you've made into a pure nitrogen environment and heat it to 500 F for a few hours. It'll carbonize.
The carbonization process creates the tiny pores which give cellulose acetate its super supercapacitor properties.
Place your new supercapacitor medium between the plates of a cap and test it out. Be gentle with it, it'll be brittle.
Alternatively, you can grind it up into a fine powder and put it into a regular capacitor liquid dielectric or a thick oil to form a thick slurry, then paint the cap plates with the slurry.
Here's some South Korean researchers who use cigarette butts to do the same: