What is the rarest naturally occurring element in Earth's crust?

Astatine, of which the entire crust is estimated to contain less than a gram at any moment. It exists only as a fleeting product in the decay chains of uranium and thorium, and its longest-lived isotope has a half-life of about eight hours - so it is continuously created and continuously gone.

Nobody has ever seen a visible quantity of it and nobody ever will. Any sample large enough to look at would vaporise itself instantly from its own radioactive heat, so its bulk properties - colour, melting point, whether it is a metal - are extrapolated rather than measured. It is the heaviest halogen, below iodine, and is predicted to be metallic, which would make the group run from a gas at the top to a metal at the bottom.

It is being investigated as a cancer treatment for exactly the property that makes it useless otherwise. Astatine-211 emits alpha particles, which are destructive over a very short range - a few cell diameters - so attached to a molecule that binds tumour cells it can kill them while sparing the tissue next door. Producing it requires a cyclotron, and world supply is measured in tiny fractions of a gram a year.