This is "Climber ‘bot". It uses three motors: one for each set of wheels and one for operating the extension. Power is delivered by a battery box, and control is done via two mechanical switches. 

I used a purely mechanical control system for two reasons: the ‘bot doesn’t have to carry the weight of a RCX, and I found it to be more challenging than just writing a program saying “Motor 1 on for 3 seconds; Motor 2 on for 60 seconds”.
As the power is turned on, the ‘bot begins extending itself towards the walls. When it reaches full extension (just short of 33 cm.), one of the switches is turned off and the extension is stopped. At the same time, another switch is turned on and starts the two motors driving the wheels. All of this is operated by a system of axles that connects the extending part of the ‘bot with the two switches. 
Notice how the two switches point in opposite directions. This is because, as one of the switches turns off the other must turn on.
I’ve taken pictures of both switches, and also have a short movie of the mechanism in action.

Upward motion is provided by a gearbox meshed directly into a system of 24 tooth gears that drives the wheels. 
The use of a gearbox serves two purposes: it trades speed for power via the use of a worm gear, and it prevents the ‘bot from sliding down since the wheels cannot turn the worm gear inside the gearbox.
I’ve taken a picture of the disassemble drive on the extension part. It is identical to the one on the non-extension part except, of course, it’s designed to slide out.

I used the small off-road wheels because they compress slightly when the ‘bot presses against the wall. This increases the contact area of the wheel and helps the ‘bot hang on to the walls. Also, these wheels are made of soft rubber and so sticks better to the walls than other wheels made from a harder compound.

The long legs are an attempt at scoring some bonus points by raising the ‘bot 16 cm. above ground. The wheels at the bottom are there to facilitate the sideway movement of the ‘bot as the left leg is mounted on the extension part.
Yeah ok, I admit it - it’s a bit of a pathetic attempt. But unfortunately I didn’t have the time to design and build a proper mechanism to lift the ‘bot. In any case, the rules didn’t say anything about not being allowed to start the ‘bot already elevated… :-)

Climb-speed is just around 50 sec. for three feet, or around 17 sec. pr. feet. 
I’ve made a movie of the ‘bot climbing. If you look to the right of the ‘bot, you’ll notice some yellow markings. The first marks 15 cm. from the ground, and the rest marks 31 cm. from the previous mark.