Those who move frequently know: The best place to get sturdy cardboard boxes for free is at a liquor store. Booze comes by the case, and if the boxes are strong enough to hold 24 bottles of Jack, they're strong enough to hold your crap. Plus liquor stores have more of these boxes than they know what to do with, and they're happy to have a dozen less boxes to break down and bale up for recycling.
Speaking of recycling, sending tattered boxes to the recycler is the best thing for them. But for boxes that still have plenty of life in them, recycling them can actually be wasteful. By subjecting them to the energy-intensive recycling process you take them out of the world before they've fulfilled their full utility.
A company called UsedCardboardBoxes.com realizes this, and has formed an entire business around getting boxes to those who need them:
I confess I have no idea why their promotional video seems targeted towards 'tweens. In any case, UCB sells all sorts of boxes—S/M/L/XL, file boxes, shipping boxes, wardrobe boxes, picture frame boxes, and the unfortunately named Gaylord boxes.
The latter type are also known as pallet boxes, but I guess when you're dealing with cardboard all day you need something to spice things up, and "Gaylord" is more fun to say than "pallet."