Excellent question!
Mountains do not grow like we might think of living organisms as growing, however mountains can change size. Mountains form through a process called orogeny, or the building of continental mountains by squeezing, crumpling, and folding Earth's crust. Mountains are formed by the movements of the vast tectonic plates that make up Earths exterior, as well as Earth's climate and processes of erosion. Mountains are changing sizes all over Earth's surface. Both mountain ranges and the peaks in them are increase in height and volume at different rates. Some mountains are rising really quickly, like the Himalayas (7 mm per year), though the Mount Everest peak in the Himalayas is only growing about 4 mm per year.

The Himalayas are forming because the Indo-Australian Plate has been actively smashing into the Eurasian Plate for 50 million years, causing long-lived crustal thickening. Other mountains are rising very slowly, like the Rocky Mountains. These mountains were formed by the subduction of Pacific oceanic crust beneath North America from 80 to 55 million years ago. The Rockies continue to rise due to buoyant forces (think marshmallow floating on hot chocolate) and erosion, but not as quickly as the Himalayas. Still other mountains are no longer rising at all, like the European Alps. These mountains formed when Africa collided with Europe about 55 million years ago. Any rise that may occur is being balanced by wind and water eroding the mountains down. Therefore, the Alps are at a steady state.

Mountains can also change depending on landslides, volcanic eruptions, gain/loss of thick ice sheets, a several other geologic processes. The rate of rise is therefore controlled by the forces causing the crust to increase in height (for example: plate tectonics, climate, erosion) relative to the forces causing the crust to decrease in height (for example: erosion, ice weighing down crust).

The speed at which mountains grow depends on rates of continental collision and erosion, as well as isostasy, which could essentially be thought of as the buoyancy of the crust. When two pieces of continental crust are pressing against each other, the crust piles up to form mountains. This is how they grow. Erosion weathers the mountains down, and makes them shorter and less steep. The important thing to realize about mountains, is that like icebergs, they are essentially floating on the mantle, and there is a lot of them beneath the surface. When the top of a mountain is eroded, the top part gets lighter, so it floats a little higher. This is what we call isostasy. Because of isostasy, mountains aren't shortened as fast as they would otherwise be by erosion. The balance of all these factors determines how fast mountains change their height. Some, like the Appalachians are getting shorter, and some grow a few millimeters or centimeters per year.

The rate at which mountains grow depends on the difference of how fast they are being pushed up and how fast they are being worn down. Mountains form and grow in several ways, some faster than others.

At the same time, erosive forces such as wind, rain, and glaciers are continually breaking up the rock and reducing the height. In some cases, erosion is occurring faster than growth and a mountain range is "growing" at a negative rate (i.e., it is getting shorter); in others, the growth is greater and the mountains are getting taller. In either case, the change is pretty small compared to the size of the mountains: the Adirondacks are growing at ~0.1 inches per year, and Mount Denali at ~0.4 inches per year. (An earthquake a few years ago cause a big loss for some peaks in the Himalayas, ~23 inches. This isn't a recurring rate though, and the Himalayas are otherwise growing. ) (Some additional explanations on mountain growth is on ScienceLine here, and here is an interesting site with a lot of information on geological processes related to mountains.)

The rate of mountain growth depends on geological factors in the region. For instance, Mount Everest grows by 4mm per year. On the other hand, some mountains have been measured to shrink! Some factors that play a role in this process include:

1) how much the tectonic plates in the region move against each other to increase the height of the mountain
2) rates of erosion
3) frequency of earthquakes, and so on.

It seems that generally, geologists quote growth rates of less than 1cm per year.

That depends on the mountain. The fastest-growing mountains that aren't volcanoes grow at rates of centimeters per year. Volcanoes that are active can grow faster if they have enough lava to erupt.

Earth's land and sea are all sitting on tectonic plates that move slowly: 10 cm/yr

Some sources state that in the North Atlantic, the rate of movement is only about 1 cm (about 0.4 in) per year, while in the Pacific it amounts to more than 4 cm (almost 2 in) annually, while others say that plates, in general, travel from 5 to 10 cm/yr:
Plate Tectonics - A Scientific Revolution - Academic Home Page.

Earth's land and sea are all sitting on tectonic plates that move slowly.