A simple circuit for a 3V high power LED can be designed using a resistor in series with the LED, calculated based on the supply voltage. For a 9V battery, a resistor of (9-3)/0.8 Ohm is needed, while for a 4.5V battery, it would be (4.5-3)/0.8 Ohm. It's crucial to use a resistor with a suitable power rating, ideally at least 10 watts, to manage the heat generated. Using multiple LEDs in series or a buck switching regulator can improve efficiency and reduce heat output significantly. Voltage regulator ICs like LM317 or L7805 are included to maintain constant brightness as battery voltage decreases.