Instead of all these special materials and boxes etc, the better and most cost effective way is to use plasterboards and build room inside room...and if its not enough dbi reduction? then put another set of boards, until you get the rating you need...
The only disadvantage of this technique is that it shrinks the space of the room. Especially if you have to place more than 4 boards on top of each other.
But if you want to make it extremely cheap, you always have the egg cartons, which don't seem like it, but even have an interesting soundproofing effect.