Almost anything that was once alive, plus paper and cardboard. The useful question is not what is technically compostable but what is compostable in your pile, at your temperature, without attracting rats. Below: the practical detail on what can you compost, and what most guides leave out.
Every entry below answers yes, no, or yes-with-a-caveat in the first sentence, then explains why. If you are looking for something not yet listed, the general rules underneath will get you there.

Was it alive, and is it small? Then it will compost. Speed is mostly a function of surface area.
Is it wet and soft? It is a green: nitrogen-rich, fast, and prone to matting.
Is it dry and structural? It is a brown: carbon-rich, slow, and what stops the pile going anaerobic.
Would it attract an animal? Meat, fish, dairy, oil and cooked food will, unless your pile runs genuinely hot. That is the real line, not some rule about what is “allowed”.
Colour does not determine category. Coffee grounds are brown and behave as a green. Fresh green leaves and dry brown leaves are different materials.
“Compostable” packaging usually means industrially compostable, 55°C sustained for weeks. Your pile does not do that. See is compostable packaging really compostable.
Composting 101 includes a one-page quick-reference of what goes in, what needs care, and what stays out. Plus the general rules for anything not on the list. Handy when you are standing over the bin with something in your hand.
Independent sources worth reading alongside this page: EPA, Composting at Home · Cornell Waste Management Institute.