I have a 12 sheet shredder (say that 5 times fast!). I'm going to give this a try. I just made a recycling run a couple weeks ago, so I may have to "save up" some stuff to get a meaningful pile of box board to work with. The shredder is just about full of office paper type stuff from shredding old bank statements.
I'll let you know how it goes.

The older I get, the more upset I get with myself for throwing stuff away without thinking if there might be a better option than the landfill.
Years ago, I would rake up all my leaves, bag them, and haul them to the landfill. Then I started to compost the leaves in my pallet bins and eventually use that leaf compost in the garden. But now, with chickens, I just dump all my leaves in the chicken run compost system and let the chickens break down the leaves into compost much faster. I'll save a few bags of dried leaves in case I need more litter for the coop in the winter.
We used to take all our tree debris from yard cleanup after a storm, or just normal prunning of branches, and load up the pickup and haul everything to the landfill. Maybe make a big burn pile of tree parts and set it on fire and just burn everything down. Since I got my chickens, I now chip up all the smaller branches and use the wood chips in my chicken coop or the chicken run. The thicker branches and tree trunks will get cut up to size and I will use them in a hügelkultur raised garden bed.
Dear Wife and I used to have maybe 4-5 bags full of household garbage to bring to the landfill every week. Most of the bulky stuff was paper products. We have a recycyle bin down the road on the way to town, so we started to separate our plastic, metal, paper, and glass. That reduced the bulk of our "garbage" leaving us fewer bags that got thrown into the dump. But now that I am shredding our paper products and using them for the chickens, and then composting the used paper shred litter, we have reduced our discarded bags for the landfill even more. Also, we keep a "chicken bucket" in the kitchen for all our kitchen scraps and leftovers to feed to the chickens. Over time, we went from about 5 bags of garbage a week down to about 1 garbage bag per week going to the dump.
Not only do I feel good about reducing our garbage we throw away, but I have found better ways to use those leaves, tree branches and parts, kitchen scraps, and now most of our paper products. I think you can see the theme that almost everything gets used in one way or another with the chickens and then, over time, ends up as compost for the garden.
I used to save up my paper products for shredding. Over time, I have started to just shred all my waste paper as it becomes available. The end result is that I don't have piles of paper or cardboard boxes laying around taking up room. Also, if you shred normal office paper, bills, etc... and then shred up some light cardboard from empty food boxes, I find I get a nice mix of paper material in the shredder bin at the start. Also, I no longer have piles of "junk paper" siting around the house to upset Dear Wife.
If my experiment with using shredded paper for the chicken coop is successful over this winter, I might do more of it in the future. If not, I'll still use the paper shreds in the chicken run and as carbon products for my pallet compost bins which I still have for overflow. There are still some foods I won't feed to the chickens, and anything moldy will get tossed in the pallet compost bins. So, I still need carbon products to balance off that stuff.