Recently KitsuneFox had mentioned that "Mount Chimborazo in Ecuador is the highest mountain on Earth, as measured from the center of the planet. The peak is over 6,800 feet further away than Mount Everest, and is the closest land on earth to outer space."
Depending on the definition of height, there are other claimants to be the highest mountain;

Mount Everest - Maximum Height Above Mean Sea level
Going by the conventional datum of Mean Sea Level (MSL) - the peak of Mount Everest is highest with an elevation of 8,848 m.
Chimborazo Volcano, Ecuador - Maximum Distance From Earth's Core
If Centre of Earth is used as datum then because of Earth's bulge at the Equator, the peak of volcano Chimborazo in Ecuador's measuring 6,384.4 km from the Earth's core to the peak is the farthest from the centre of Earth.
It is therefore highest point on Earth and is closest to the moon, stars, and space.
Using this datum, several other Andean peaks and also Africa's highest mountain, Kilimanjaro, are higher that Everest, which is about 2.1 km closer to the Centre than Chimborazo. However MSL elevation wise Chimborazo is much lower having an elevation of only 6,267 m.
Mauna Kea, Hawaii - Maximum Height From Bottom of Sea Floor
If distance from the bottom of the nearby ocean floor (or base) to the peak is used as a datum then the peak of Mauna Kea, a dormant volcano in Hawaii, will be higher than Mount Everest. It has a height of 10,203 m as compared to 8,848 m of
Mount Everest. Again going by the conventional MSL its elevation is only 4,207 m.
It may also be worthwhile to note that Mount Everest is often treated as "True Mountain" as it was created by collision of two tectonic plates where as Chimborazo and Mauna Kea are volcanic peaks.
Source: geology.com / forbes.com