Very good answer. Mauna Kea is in fact the highest mountian on Earth with much of it below sea level. It is around a kilometer higher than Everest. Everest is the well known mountain as it is the highest above sea level and the one that gets climbed a lot.
However, this is not the highest known mountain.
Well done JonB. The highest known mountain is Olympus Mons. Way higher than anything we have locally, it is on Mars. You will find some photos and info here
http://www.olympusmons.com/