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Do you use the door at all? If not, multiple layers of cardboard, or bubble wrap if you want some light, should give you an R-value not too much less than standard insulation of the same thickness. As was mentioned above, it's the trapped air that does most of the insulating, and corrugated cardboard should do that pretty well.
I'd try to run the corrugations horizontally, rather than vertically, to minimize any convection.
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