Most people's immediate reaction to that question would be: the top of Mount Everest. But that would not be correct.
Mount Everest's 8,849 meter (29,032 ft) peak places it as the highest above sea level. If you measure from base to peak, however, Mauna Kea in Hawai'i is the tallest on Earth, measuring over 10,210 meters (33,500 ft), despite only peeking out just 4,205 meters (13,796 ft) above sea level.
However, to get the furthest from the centre of the Earth, you would need to scale Mount Chimborazo in Ecuador, which is only the 39th tallest mountain in the Andes. It also happens to be the spot on Earth that's closest to the stars. How so?
Well, it's all to do with the fact that Earth is not exactly round. As the planet spins, the centrifugal force causes the planet to bulge at the centre by about 27 miles (43 km) - which just happens to be along the line where Mount Chimborazo sits. With this (slightly) unfair advantage, the peak of Ecuador's Mount Chimborazo is the furthest point on our planet from the centre of the Earth, standing 2,072 meters (6,800 ft) further from that centre point than Everest's peak.
If you stand atop Mount Chimborazo, you are also the closest you can be to the stars without leaving the ground.