A machine and method is disclosed for stacking product, including but not limited to, cardboard, magazines, newspapers, newspaper supplements, flyers, press signatures, and softcover books. The stacking machine includes an infeed assembly for receiving a product stream or flow. The infeed assembly includes a belt jogger assembly for aligning product as it enters the machine. A transport assembly delivers product from the infeed assembly to a stacking assembly. The stacking assembly includes an interrupter that creates a gap in the product stream and assists in accumulating a single lift of product on a separator. The separator delivers a single lift of product to one of a pair of elevators, which deliver a first application of compression to the lift. After the first compression, the lift is transferred to a compression apparatus where a second round of compression is applied in the process of forming a final stack. Each time a lift is delivered to the compression apparatus, compression is applied to the stack, thus many of the lifts will be compressed several times until the stack is ejected from the machine.