While special three-way bulbs can be used for touch lamps, they are not actually necessary. Because of the way touch-sensitive lamps work, the brightness of the bulb is controlled by the amount of power being passed to it when the base is touched.
Most touch lamps use three separate brightness settings that allow the user to adjust the lighting as needed. When the user touches the lamp, a specific amount of power is passed through to the bulb, in effect changing the duty cycle of the electricity. While bulbs in normal switch lamps automatically get full power, touch lamps limit this by varying degrees to enable multiple brightness settings.