On-topic: I've read something about a mile-high skyscraper too. I can't be bothered to find it at the moment, but that'll certainly be an interesting project too.
Well I had seen that in a documentary about a huge tower built in China.
I can't explain it very well but basically the crane is stabilized by multiple hydraulic ''legs''. When the time comes to go higher some of these ''legs'' go to the higher floors, and they lift the crane up going back to the base of the crane.
Think of this like you getting on a high wall. First you'll extend your arms, then with your own force you will bring your feet to the height of your hands and the wall. Once on the wall you'll be able to stand up.
If anyone can offer an even more laic explanation, I salute you!
Mega-projects are one of my fave things - just been looking at the Wiki and it seems that there is likely to be a 1,000m (1KM) high building likely to be made in Dubai, however anything over this is usually just fancy (i.e. the "The Illinois" which was to be 1 mile high, but had major design issues).