Where do you think you need to go on Earth, to be as far away from the centre of the Earth as possible?
If you said Mt Everest, you'd be wrong. It's a peak called Chimborazo in Ecuador.
Map of the Distance to the centre of the Earth.
The map here shows the terrain relief. But rather than showing heights relative to sea level, it shows the distance from the centre of the Earth.
As you can see, the furthest 10 peaks from the centre of the Earth are in Ecuador and Peru - along with Mt Kilimanjaro.
How does this work?
This doesn't make intuitive sense.
The reason is the Earth isn't round. It's not a perfect sphere. It's wider round the middle. The technical term is oblate sphereoid.
The diameter of the Earth is 22km wider round the Equator than it is around the line going through the two Poles. The Earth is squished vertically - only by 0.3%, so it's not perceptible when you look at it. This slight bulge means that high mountains near the equator get pushed further away from the centre of the Earth. Mountains closer to the Poles (such as Mount Erberus in Antartica) move closer to the Earth's centre.
Mt Everest is around 8km high. Less than the amount of bulge. And it's at 27 degrees north, so it doesn't get much advantage from being close to the equator.
Likewise, the points closest to the centre of the Earth are not in the Mariana Trench; instead, they're in the Amundsen Basin near the North Pole.
There's another factor in play too... the geoid [gee-oyd]. There are changes in gravity across the surface of the Earth, caused by factors such as the surface thickness of the crust. This means that the distance from the centre of the Earth to sea level varies across the globe, by as much as 200m (+/-100m around mean)
Steven wrote a Python script (with GDAL and numpy) to merge the data from ETOPO and the GEOID and apply the calculations shown in the first citation.
SAGA was used to find the peaks, and QGIS used to sort these by height and discard all but the top 10 highest ones.
QGIS was used to render the map.
 Based on calculations detailed here: http://www.summitpost.org/distance-to-the-centre-of-the-earth/849764
 Geoid Data (EGM2008) http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/
 ETOPO Global Relief Model https://www.ngdc.noaa.gov/mgg/global/global.html