I regularly throw in stuff like toilet paper rolls/paper towel rolls (cardboard), scraps of notebook paper (better than the trash, and closer than the recycle bin when I'm in the kitchen) or other paper/cardboard products. Seems to work ok.
Straw is good, but watch out for seeds.
small sticks (smaller than pencil) decompose slowly and usually have leftover chunklets in the final product, but add structure to the pile and are mostly carbon.
Used lightly, some fireplace/bonfire/bbq ash doesn't seem to hurt anything, and it is a convenient way to smother stinky rotting veggies.
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