I am not a person who saves my packaging from my embellishments. It would be great if I was and I have started to try to be more deliberate about that...but I definitely haven't before the last 6 months.
I just simply use a plain white plastic drawer like one that comes in a storage compartment. Now, with letters in it, it does not fit in as a drawer. But that's okay because my drawer part got broken by one of my kiddos! And honestly...I kinda like it being open and just sitting on a shelf! My container measures about 10 inches wide by 14 inches long. I only have one drawer because I am a purger and don't have any more Thickers than that! I know that if you got a little bigger drawer (taller) you could make it work in a shelf system and have several drawers full of Thickers if you have lots!
Here you can see a side view of how they fit in. If I have kept the packaging, I trim off the top label part but keep the celophane around it (slicing open one long side and one short side for easy access but avoiding the stickyness at the bottom)
If I don't have the packaging anymore then I put them together facing it's partner...preventing it from rubbing on other and getting lost.
It's a very simple system and has worked great...I just wish I had thought of it sooner ;)