If you have miles of steel pipes 10m thick, could you pump water from the sea to the centre of Antarctica, then pump it into the air like a geyser so that it falls around the pipe. As the fallen seawater freezes and makes a cone, extensions are added to the pipe to raise the height and more water is pumped through. Every kilometre, a pumping station is added, until the pipes and the frozen cone grows higher. Can such a mountain be made tall enough to allow a train running on a train track going up the cone to escape earth's orbit?
The maximum height a mountain can have on Earth is a tad more than what Mount Everest is high. This is due to the fact that when you increase the height of the structure, you are also increasing the load. After a certain point you will be adding too much weight for what the material can sustain, and the entire structure will crumble on itself. The potential energy you will be storing in the structure will be higher than the energy needed to melt it, et voilá.
Incidentally, this is also the reason why planets are spherically shaped.
Now, ice is even less resistant than stone, so it doesn't take long to answer no to your question. Moreover, as all the glaciers around the world show, when loaded, ice exhibits a plastic behavior and flows. Another reason for making what you ask impossible.
You've already accepted an answer, but aside from the structural issues (and the sheer mindboggling amount of energy it would take to pump all that water up that high), there's another misunderstanding:
allow a train running on a train track going up the cone to escape earth's orbit?
Getting into orbit isn't simply a matter of getting up really high. The altitude at which you orbit is strongly related to the speed you orbit at (and vice versa); so the most important thing is to be moving really fast. A mountain just high enough to poke into the Low Earth Orbit region needs to be about 160 km high, and at that altitude you need to be going over 7.8 km/s for a circular orbit or you'll just fall back down to earth. The fastest trains in the world are around 267 mph (or ~165 m/s), so they're clearly never going to be able to enter orbit.
If you want to just "step off" your hill into orbit, you'll need to be all the way up at geosynchronous altitude... 35786 km. That's over 5 times the radius of the earth, so you'll have a problem finding enough water in the solar system to build an ice mountain that high. (edit: also, as people have pointed out, the mountain would need to be at the equator so earth's own rotation would provide orbital speed at the top of the mountain, so your polar arrangement would always require a substantial speed boost of the order of several km/s, give or take an km/s or two)
If you allow any sort of train, then you can build an electromagnetic launch system, like a Star Tram. The biggest star tram design, the gen2, needs at least a 1000 km of magnetic track, inside a vacuum tube, with the business end 22 km up in the air. The design has the advantage that it doesn't even need a huge hill to support the top end, and so is easy to assemble compared to trying to build a mountain into space.
And this is without going into the details of circularisation burns for orbital insertion... getting into orbit isn't a trivial exercise!
L.Dutch has already explained why it won't work, I want to add that if you try to do this in any reasonably short amount of time, you will create a huge turmoil.
- The amount of energy needed to pump thousands of thousands of cubic kilometers of water 3 km uphill (and that's just in the beginning) is
- A good part of this energy (+ the heat of crystallisation of the water(!)) lands in the center of antarctica, making a huge low pressure weather system, in other words: a cyclone.
- this draws in more water from the air around Antarctica, which precipitates on your mountain (so far so good!)
- the increasing pressure speeds up the glaciers, Antarctica grows
- the inflowing air from the north is relativeley warmer, much more turbulent weather
- you move mass closer to earth's axis, and the conservation of angular momentum dictates that earth's speed of rotation increases
- if you don't do the same on the north pole, things might get a bit wobbly
- before your mountain has reached ten kilometers in height, the sea level will have sunk by twenty meters or so.
- then, Antarctica gets so heavy that it starts to sink (like the continents did in the ice ages, only faster), and the oceans swallow up large parts of it.
- and what all that will do to earth's tectonics, I don't want to imagine.
All in all, I'll say this is a bad idea. ;-)
Starfish Prime's answer touches on this, but all the answers so far seem to be missing the most significant problem with this question.
The question assumes that it is possible to "escape Earth's gravity". This is a very common misconception, based on the observation that people in the space station float around as if there is no gravity. The "as if" is the important thing. The British TV series "Rocket Man", was based on this false concept, so the questioner is in good company.
In fact the Earth's gravity is still very much in effect in the space station, the force being 85% of what it is at the surface of the Earth. The astronauts and the space station itself are all affected by this gravity and are continuously falling and accelerating downward toward the Earth. They appear to have no gravity because they are in free-fall, as are the observers.
What we think of as gravity is actually not gravity itself, but the effect of resisting gravity. Gravity pulls us down, but the floor prevents us from moving. This pressure we feel on our feet is the force that resists gravity, not gravity itself. If we stepped off a high cliff, that feeling would go away. We would feel weightless, as if there were no gravity, yet we would be very much affected by gravity in that situation.
What happens is that as the astronauts fall down toward the Earth, they are also traveling parallel to the Earth at 27,600 km/h (17,100 mph). So as they fall, they also move sideways to a part of the Earth that, due to its curvature, is farther away. The sideways motion, the gravity, and the curvature of the Earth all balance out and the station ends up traveling in a circle, orbiting the Earth.
Anything at that height traveling sideways at a faster speed will move away from the Earth. Anything traveling at a slower speed will move toward the Earth. But when the orbital speed and height are perfectly matched, the station stays at the same distance from the Earth's surface.
Imagine being fired out of a cannon aimed horizontally. You would travel in a path that slowly, and then more quickly, curves down to the ground. Now use a more powerful cannon and you will still follow a curve, but you will travel farther before you hit land. Keep increasing the power, and eventually your path will curve down at the same rate that the Earth's surface curves down. You'll go so far that you'll actually go all the way around the world and crash into the back of your cannon. This process is how the space station stays in orbit. (The cannon scenario ignores air resistance and the Coriolis effect, but where the space station is there is no air, so that's not a problem.)
Really, the concept that "there is no gravity in space" is totally wrong.
So the basic premise of this question is false. The stated purpose is impossible. It's also an excellent example of "The X Y Problem".
This would be a much more realistic question (actually two) if it were asked as:
How high could an ice mountain be built on Antarctica?
If we could build a 100 km (60 mile) high platform, could we use it to put satellites into orbit by firing them from a cannon or using an electric sled to accelerate them sideways to 28,000 km/h (17,500 mph)?
There are potential projects that aim the same thing. Getting to the orbit without rockets by using today's technology. Obviously none of them are using a giant ice mountain. This may provide some insight to these methods: https://en.wikipedia.org/wiki/Non-rocket_spacelaunch