I take all the long pieces and braid them together and then add several knots to make simple dog toys. All the long seams, the plackets, etc. (I remove the buttons first, of course) They don't last forever but they do seem to last at least 8-10 months (they mostly play keep-away with them) and it seems like I always have more leftovers ready to go when it's time to toss one out. My dogs get SO excited about those toys, it's really cute how thrilled they are when I bring a new one out.
Anything too small usually just goes into my scrap bag, which eventually turns into a dog or cat bed.
I guess my pets get 99% of my scraps, LOL.