Packing paper is one of my cat's favorite things to play with and sleep on. I highly recommend keeping the packing paper from your next delivery.
Luke loves to hang out under the paper and plot an attack.
With each delivery of Chewy comes a huge piece (or pieces) of large brown packing paper. As soon as it comes out of the box and I start unfolding it, the cats come running. I layer it either by itself, or fold it on top of a tunnel. The cats can run into it, under it, and over it for hours. It's crinkly and fun. It's a great spot to play with toys, or even just hang out on.
My cats also love to sleep on it. Another option is to cut the paper into large strips and fold it for sleeping. I put it into cardboard boxes and it's a huge attraction for resting on.
So, don't throw away that packing paper - let your cats enjoy it.