Do we get a prize for finding all of the gold chests on each planet? I have yet to do that.

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You get an upgraded Sparrow for finding all the gold chests in Old Russia (it will be in the last chest you open). One of them seems to always give a rocket launcher as well (the one in the Dark Within). I never got anything memorable from any other ones, but I did get weapons/armor, so I suspect those were the prizes.

There doesn't seem to be one really big prize for finding all of them on all the planets, though.

Your prize is the warm glow of a sense of accomplishment. Or if you're the obsessive completionist type, the satisfaction of filling up all the number bars.

Go forth and prosper.