Operate Car Stereo without ignition? Battery Drainage?

Neither a capacitor nor an amplifier will do anything to let you listen to the system longer with the ignition off. The one thing that can help is a second battery. You can install a battery with an isolator so that when the engine is off, your system only draws power from the second battery, leaving the primary battery fully charged so you can start your engine no matter how long you've been listening.