What term is used for a star that has exhausted its nuclear fuel and is very dense?

The term used for a star that has exhausted its nuclear fuel and is very dense is a neutron star. A neutron star is the remnant core of a massive star that has gone supernova, and it is incredibly dense due to the collapse of the star's core under gravity, leading to a state where protons and electrons combine to form neutrons. This results in an object with a mass greater than that of our Sun, but compressed into a sphere just a few kilometers in diameter.

In contrast, while a red giant refers to a late phase in the life cycle of a star before it may shed its outer layers, a white dwarf is a less dense stellar remnant that forms from stars of lower mass and will not necessarily reach the density of a neutron star. A black hole, on the other hand, represents a stage of stellar evolution where gravity is so intense that not even light can escape, and it is an end state following the neutron star phase for the most massive stars. Thus, neutron stars are characterized specifically by their extreme density, out of the given terms.