Everest is the highest mountain on Earth, 8,849 metres above sea level, so naturally it is also the closest bit of solid ground to outer space. It is the obvious conclusion, and it is the kind of thing you would bet money on at trivia.
It depends what you measure from. Everest is the highest above sea level, true. But the Earth is not a perfect ball; it spins, so it bulges out at the equator by about 21 kilometres. 'Closest to space' means farthest from the centre of the Earth, and that prize goes to Chimborazo, a volcano in Ecuador sitting almost on the equator. Chimborazo's summit is only about 6,263 metres above sea level, a good two kilometres shorter than Everest, yet because it rides the equatorial bulge it pokes roughly 2,000 metres farther from the planet's core, and closer to the stars. NOAA spells this out plainly. Everest keeps the 'highest' title. Chimborazo quietly takes 'closest to space', and almost nobody has heard of it.