We like to pile it up, and then go chase it down the driveway when the wind picks up, then pile it up again.
Just kidding. I don't love chasing cardboard. But I do love having a pile of it. My plans for the cardboard pile include: expanding gardens for next year, covering the lawn as I begin to build my small forest, and making toad habitat.
Last year, we weeded out a terribly overgrown side of the house to build a shed. Around it, we lay cardboard for many months, and this spring, we put in a nice layer of mulch for a path around and behind the shed.
Another favourite use of cardboard for me is building (extremely) rough prototypes of furniture I invent, like a desk or shelves. I like to be able to see how the piece will work in 3D before I take the time to build it.
EDITED TO ADD: oops, I've just seen this is posted in fungi. I guess my suggestions are not of the type you're looking for! Sorry. Bonus suggestions, I guess. :)
Bring me the box labeled "thinking cap" ... and then read this tiny ad: