Okay, I'm not a geologist but this is a rather loaded question.  It depends on your definition of 'highest mountain.' 

There is one mountain in Chile that projects further into space from the Earth's centre due to being on the equator.  There there is the Hawaiian volcano that starts at the bottom of the ocean and is actually quite a bit taller than Mr. Everest.  Or are these just urban legends?

Beyond that, the continents have broken up, moved around, and smashed into each other repeatedly, raising mountains that have then been worn down by ice ages, rain, etc. 

The fact is that even Mt. Everest is not a 'finished' mountain.  It is still rising, as are the Alps, the Rockies, etc.  But they all have ice on the summits and ice expands and contracts, breaking down the peaks as they are rising.  Who knows how tall Mt. Everest would be if it had no weatherÉ

I read somewhere that the Scottish Highlands or the Snowdonia mountains would originally have been as tall as the Himalayas but they have become eroded. Mechanical considerations (pressure, density, etc) limit the possible height of a mountain. Again, I read somewhere that there could not be on Earth a mountain as high as Olympus Mons on Mars.

Olympus Mons on Mars is so much larger than any other mountain on Earth because Mars has no plate tectonics. It has roughly the same amount of volcanic rock than the Hawaiian Island chain but because the Pacific tectonic plate is moving north-west over the hotspot the mass of the magma ejected is spread out in a line. On Mars all the magma is erupted in the same place building the volcano larger and larger. I cannot remember exact values but it should be quite easy to find. Kilauea

Mt Kilauea in Hawaii is the tallest mountain from base to peak.

It is possible that there were higher mountain ranges in the past. There is no evidence I can think of for how high ancient mountain ranges stood but I know that we can measure the mass of eroded material. For example, The Himalaya are being eroded as they are growing. The eroded material travels down rivers to the sea and is deposited in the Bengal and Indus Fans in the Indian Ocean. If you search these and see them on a map you'll see they are massive, giving an indication of just how much of the mass of the mountain has been eroded.