I have many cardboard boxes of odd sizes for my two computer monitors, speakers, laptop, coffee maker, steamer, sampler, printer etc. because I usually move every year and like to keep the packaging so that my stuff has less of a chance of breaking during moving. Are there any uses that don't damage the integrity of the box, or any good way to keep them organized and out of the way? They just continue to fall out of my closet and can never seem to stay put. I plan to live in apartments until I have enough money saved to buy a house in cash so I won't have a basement, spare room, or that kind of storage place for a long time. Also, I have kept all of the packaging like styrofoam and cardboard inserts because it keeps the appliances more secure during moves.
Unless you have kept all of the packaging with the boxes, such as styrofoam inserts and other such pieces to keep your stuff completely immobile, there is no good reason to keep the boxes. Go to your local grocery store and ask for boxes. If you go to a big store like Walmart at night and talk with a stocker, they get rid of hundreds of boxes a night and are usually totally willing to give them to you.
If it is possible to open the upper part of the box and the bottom without damaging the box, then you can fold the box and make it flat. When you need the box unfold and close the bottom and the upper part. The styrofoam you can put in some of the other boxes. This way you will not reduce the volume of all the boxes but only a part of them. The overall result will be decreased volume of some of the boxes.
You can get rid of the styrofoam for some of the appliances which are more durable and for which is possible the protect them using "bubble wrap" and old newspapers.