A further detail to the information in the how stuff works article on cranes - the example shown is of a top crawler crane - the type of crane which adds incremental sections to the top, while continuing to extend all the way to the ground. Top crawler cranes are used in many jurisdictions, including London, where there was a major crane accident in Canary Wharf, when adding a section to the top of a crane approximately ten years ago, and the entire crane fell down. In Toronto, bottom crawler cranes are generally used for taller buildings. After the initial installation of the crane on a concrete base at the bottom of the excavation, when the building has reached a point where the crane needs to be raised, hydrolic jacks are used to slowly lift the base of the crane up to the next level. The jacks are then raised up to position them for the next lift of the crane. I think the base of the crane extends about four or five floors into the building at any given time.
When the time comes for the tower crane to be removed, if the job is within range of a mobile crane, after lifting all the crane pieces down, the last hoists from the mobile crane would be lift up the cement bucket loads needed to patch the hole in the roof where the tower crane used to be.
AHK