Katie has given some excellent advise! For me, I would use 1/4" hardwood for Steve's veteran plaques. For crisp details, I would stack only two layers of 1/4".
If you are going to apply the piece to a thicker board, then you can use 1/8" Baltic Birch plywood. I wouldn't go any thicker that four layers of 1/8".
To stack the boards, I would use either clear packing tape or a hot glue gun to glue just the edges of the stack.