Tectonic plates created the Himalayas around 40 - 50 million years ago, and these plates continue to cause a general growth in then height of the mountains. However, this doesn't explain why Everest is so much taller than its neighbouring mountains. It is 250m taller than the second highest mountain in the Himalayas, yet only 120m stands between Everest and the second and third highest mountains in the world.
In a study led by Dr Xu Han of China University of Geosciences, and co written by Matthew Fox and Adam Smith, they concluded that a river 47 miles from Everest was the catalyst for its growth. The rivers power was exacerbated 89,000 years ago when the Arun river merged with the Kosi River. River Arun is a steep river which has created a gorge over thousands of years, and consequently considerable amounts of erosion. Now while we might typically expect erosion to make the mountain shorter, a scientific process called 'uplift' or 'isostatic rebound' occurs to do just the opposite.
When a section of the earths crust loses its mass, as it does through the erosion from River Arun, pressure from the mantle (the layer below the crust) causes the mass to 'float' upwards; 'uplifting' it.
Adam Smith compared this process to a ship in an interview with the BBC; “It’s a bit like throwing a load of cargo off a ship," "the ship becomes lighter and so floats a little higher."
While this process is incredibly slow, and Everest is only growing around 2mm annually, in geology timelines; this is impressive. In the past 89,000 years, its height is estimated to have grown between 15 - 50 metres.
If you're thinking of climbing Everest - I'd recommend pursuing this sooner rather than later. You don't want to risk having to climb the extra few millimetres...
You can read the study here.