Why do chicken lay eggs with shells made of lime although they never eat anything containing lime or live in?

Lime is calcium carbonate, which is what we use to make bone. It is available from a wide variety of plants.

Calcium doesn't just make bones, it is essential for other biological processes. This is why you can lose bone mass - if you get low on calcium, the body will leech it from the bones for more important functions.

In chicken eggs, not only does the shell protect the embryo, the calcium is leeched from the shell to provide calcium for the growing chick. By the time the chick is ready to hatch, the shell is thinned down enough for the chick to get through.