Mount Everest is the standard answer for highest spot in the world because it rises 29,028 feet (about 5.5 miles) above sea level. Sea level is the base from which everything is measured.
But if one measures from the actual base Hawaii's Mauna Kea is king at 33,476 feet (more than 6.33 miles) from base on the Pacific Ocean floor to tip. However, measured from sea level Manua Kea is only 13,796 feet tall.
Chimborazo in Ecuador comes in at only 20,565 feet but it has geography on its side. The earth is not a perfect sphere but instead one ugly thing called a geoid. Besides a depression at the southern pole a trait of the geoid is a bulge around the equator. While Chimborazo is some 8,463 feet shorter than Everest the bulge of the earth makes it closer to the moon and stars.
So scratch your heads and debate the meaning of tallest/highest and decide which truely is the tallest/highest mountain on earth. But remember, Mount Olympus (Olympus Mons) on Mars is the champion of the solar system and known universe rising to a mammoth height of 88,580 feet or 16.7 miles above the surface!