The elevator design your referring to has two moving stages... and believe it or not, you can power both stages with a single motor! We built a similar elevator in 2008, and it looks like the team is leaning that direction again.
http://www.chiefdelphi.com/media/photos/30267
http://www.chiefdelphi.com/media/photos/35975
http://www.chiefdelphi.com/media/photos/35974
In order to power it with just one motor, you need to think through how the pulley's and the cable that lifts it moves. It starts near the bottom, and runs up to a pulley on a stationary cross member (this pulley never moves). From there, it runs down to a pulley on the outer-most
moving stage. If you were to just tie it off here, it would life that stage up. But, since you go around a pulley there, you can send that up to a pulley at the top of the same stage. The distance between these two pulleys is fixed, since they are part of the same stage. From there, you run it down to the inner-most moving stage, and tie it off.
When you end up powering the motor, the inner-most (smallest) stage moves first. When it gets to the top, then the middle (the outermost moving) stage moves. In the end, the whole thing extends!
Attached is a quick diagram of how the cable runs work. The blue is the cable, and the gray is each stage. White circles for the pulleys, a big black circle where the cable gets tied off.