The Farthest Point from the Centre of the Earth
I'm afraid that the guess of Mount Everest* would trip the QI klaxon. The Earth bulges out at the equator due to its rotation. As planets slow down they become less oblate (elipsode-ish) and more spherical, but at present someone standing on the equator would be 21 km farther from the centre of the Earth than someone standing at the pole. Eventually this radius size difference will shrink away as has already happened with Mercury and Venus. Here's a table of the size of the larger planets in the solar system and their bulges relative to their size (given as diameters rather than radii).
|Planet||Equatorial Diameter||Polar Diameter||Difference||Flattening Ratio|
|Earth||12,756 km||12,714 km||43 km||1:299|
|Mars||6,805 km||6,755 km||50 km||1:136|
|Jupiter||143,884 km||133,709 km||10,175 km||1:14|
|Saturn||120,536 km||108,728 km||11,808 km||1:10|
|Uranus||51,118 km||49,946 km||1,172 km||1:44|
|Neptune||49,528 km||48,682 km||846 km||1:59|
Notice how disc like Saturn is! Just to be clear, that isn't the rings, it is the actual main body of the planet. In comparison the Earth is really spherical although it is oblate enough to have the furthest point from the centre be not the highest point above sea level. The honour of farthest from the centre belongs to Mount Chimborazo in the Andes, which although it only stands at 6.267 km above sea level, (only three quarters of the height of Everest) manages to come in at 6,384.4 km from the centre (which is essentially just the 21 km of extra height from being almost at the equator plus its 6 km of height).
The next farthest point is only 10 m less from the centre and it is called Mount Huascarán which is also in the Andes. This figure is so small that we can't say for sure which mountain actually holds the record and we can get into debates about what constitutes the centre of the Earth. For instance, the centre of mass is not the same as the centre of the "circle" which comprises the equator. However I can't find any accurate data on this.
Going down the list of these farthest points you have to go to number six until you get to the first recognisable name: Kilimanjaro stands at 6,384.0 km, so a good 400 m short.
*Bonus fact about Everest: it was named after Colonel Sir George Everest, a Welsh explorer and surveyor of India and Nepal. He pounced his name Eve-er-est (like Adam and Eve) rather than Ev-ver-est.