I have all my boxes and all styrofoam inserts. I might have cleared out 1/4th of the peanuts, but I have all the rest.
I recommend keeping the boxes. Even if you throw away the styrofoam and peanuts, if you decide to move or sell you can always buy packing, but not original boxes. HL sells boxes but not original ones and if you did buy then it could get pricey. Also packing is expensive also, And with the price of oil going up so will the price of styrofoam(a petroleum product). I personally feel the styrofoam will eventually be replaced with paper products.

If you do get rid of the boxes I recommend selling them. I could always use some of the old blue types.