Farthest point from the center of the Earth
It's Chimborazo, Ecuador, but only just, beating Huascarán, Peru, by less than 50 metres. Both are over 2 km 'higher' than Everest.

I made a plot of some mountains — height above centre of the earth vs absolute latitude. You can download the IPython Notebook source code here. Warning: v. hacky.



I can't find anything on the position of the centre of the earth. The formula I used for the latitude-dependent radius requires major (equatorial) and minor (polar) radii, but I don't have citations for them either. Argus in his article Defining the translational velocity of the reference frame of Earth gave some numbers for its temporal variance, but I have no idea how this might affect these mountain heights. 

Last thing: Apparently, the floor of the Arctic Ocean is the closest point on the surface to the Earth's center (about 6353 km, 30 km 'below' Chimborazo), if you call the bottom of the sea the 'surface'.