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I believe the term for height from base is called prominence. The highest one can climb is Denali in Alaska, at around 5500m. Obviously omitting mountains that have their bases on the ocean floor like Mauna Kea ofc.
Prominence is a bit more complicated than that. It refers to the height of a mountain or hill's summit relative to the lowest contour line encircling it but containing no higher summit within it. Everest's prominence is actually still its height above sea level.
Everest is a bit unique in that regard since, as the highest peak, by definition it can't have any higher peaks within any contour around it.