I can speak for the serious altitudes of the Andes, where I live. Our ozone layer is thicker than over any other high elevation (yay, equator!) but we're pushing into the middle layers of the troposphere, where there is less moisture (since the mountains tend to strip the clouds, as Dillbert mentions) and less oxygen. Our atmospheric pressure is also greatly reduced (I experience swelling when I travel from the coast to Ambato - there's literally less holding me in up here).
What it means for gardeing is threefold: first of all, because of reduced atmosphere (I live about 3 km up) the sunlight hits us faster than it does at sea level, and there's less atmosphere actually there to impede the light and heat. This means that daytime temperatures can be extreme, even mitigated by the fact that it's cooler up this high. I can have 40 C days easily (it's shaping up to be one now!). However, that same effect works against us at night, meaning that it cools off quickly and fairly drastically - 5-10 C is a normal overnight. This means that whatever you grow has to be both heat and cold resistant. The plants that are found naturally in high-altitude environments, cacti included, often have white or silver fur (look up Espeletia, an Ecuadorian endemic, for a neat example) to act as natural sunscreen and insulation. Plants that don't naturally have that, like say tomatoes, need special treatment in order to deal with the sun's effects. Shadecloth and greenhouses are used in my area, which is where some 90% of all tomatoes for the country are grown - they die in full sun. You also get the odd effect I mentioned where water droplets on leaves can cause focussed sunburn, which means that most irrigation up here is by drip, rather than overhead.
The Andes in particular also generate their own unique wind systems, and in my part of the chain at least, that's driven by the volcanoes and the shape of the mountains. When the "seasons" change, we can get extreme winds of over 100 kph, and that means that whatever you're growing also has to be wind resistant, or you have to figure out a way to support it when it does get blowing. Volcanic eruption wreaks havoc at high altitudes, as the quantities of ash produced are enough to give us winter weather (cold, wet) even in the deserts.
Thirdly, at very high altitudes below the snowline, it either rains all the time (wet paramo) or it doesn't hardly rain at all (altitude desert - where I live). Depending on where you are, that means you've got to ensure insane drainage or grow xeric-friendly plants and irrigate like a madwoman. Drip systems are the only really good option in the deserts, and in the wet paramo people generally grow water-hungry and cold-tolerant crops, like squash and beans.
There are some things I can't grow at all up here in the rare air - pineapples are grumpy and refuse to fruit (even in the greenhouse, so it's not a temperature issue) and there are numerous fruit trees that can't stand the cold overnight temperatures. However, in terms of oxygen, there's nothing doing - if it will grow for me up here, it's not because the plant is more tolerant of rarified air (plants produce O2, remember?) but because it's more tolerant of the extreme conditions. And the altitude actually works for me - I'm in the only location on the equator where stone fruit can be grown, because we get cool enough nighttimes for the trees to be comfy. That means I've got plums and peaches in my yard (blooming right now, actually), something that I couldn't really do anywhere else in the country - my province is famous for its stone fruits, and my city is sometimes called The Big Peach.