How do they erect HUGE cranes without a crane?
The really huge cranes with the boom that goes straight out at a right angle to the upright section (which goes straight up) is called a tower crane. They're assembled by using a jack assembly at the bottom, so that more sections are stuck in at the ground level.
They use Hydraulics. I don't know if you have any understanding of Fluid power but a hydraulics system can lift up to 400% of its weight multiplyed by its density. There are other factors to consider such as cylinder size, flow, and pressure but these are usualy constants. For example: If the hydraulic fluid used weighs 5 lbs. per gallon and its density is .9 gram per cubic centimeter then 1 gallon of this fluid could lift 53.1441 lbs. A jack system is hydraulic