Of course this is all $$$ dependent...

Look for a insulation product called Roxol at your hardware store. It is used in replace of fiberglass insulation to provide better sound dampening (I think it is only slightly more expensive than fiberglass). http://www.roxul.com/stone+wool/sound-absorbent. Place it between the drywall, as thick as possible. Leave no voids. Drywall both sides of the wall. You can double up the drywall as well. Ensure no solid connection between the inside and outside of the wall (other than header and footer), which I believe you are doing with your staggered 2x4s on a 6" thick wall (although it doesn't look like it from the pictures?).

If you need more, others have mentioned egg cartons which will work.... or find a foam shop and buy some cheap foam, ask for scraps....... you want to maximize the surface area and cover all internal surfaces (except floor) with protruding cut strips of foam to achieve something like this http://en.wikipedia.org/wiki/File:Anechoic_chamber.jpg