What are the non-cellular structures in animals?

Megascopic structures that simply grew by accumulation/precipitation? I can only think of teeth enamel (or maybe I’m wrong there) and nothing else. I know skin epidermis and nails are dead skin cells. Anything?

A cell’s cytoskeleton is the scaffolding that holds the cell’s structure but is not itself made of cells.

The general term you’re looking for is “extracellular matrix”. Bone, cartilage, and connective tissue in general are not primarily composed of cells. Rather, cells secrete the various extracellular matrix proteins and minerals. There’s plenty of other extracellular matrix everywhere else, but not as abundant as in connective tissue.

Blood is another tissue with more extracellular than cellular material, but you probably wouldn’t call it a ‘structure’.

Elsewhere in the animal kingdom you can find a wide variety of shells and cuticles that don’t contain many cells. And there’s mesoglea, the jelly that makes up a jellyfish’s skeleton.

The most abundant tissue in most animals is skeletal muscle, and at least in vertebrates this is a syncytium: it has nuclei but they’re not separated by cell membranes. So in a sense it’s non-cellular, although you can also regard each fiber as one very large multinucleate cell.

There’s the vitreous humor as well.