If you're willing to buy and use a great number of nine-volt batteries and a stereo-to-mono quarter-inch cable, you can get along without the power supply, but I wouldn't recommend it.
Series basses run at 36 volts, and the power supply allows you to choose between running both pickups/preamps summed into one channel of your amplifier, or running the bass and treble pickups seperately into different channels or even different amplifiers.
For example, I run my Series 2 into one channel of my stereo Walter Woods amplifier with great result, but if I wanted to get fancy I could run each pickup into a seperate channel.