After getting some ideas off this board, I purchased 5x8 mini binders and heavy weight sleeves from Keepfiling after some Googling. The quality of the binders (~$5) are fantastic and I find them really attractive (black with green/blue/red/white insides, or white all around). I wanted to avoid the generic ones sold on Amazon since they looked a bit cheap and was not going to shell out for the official Playbill ones. They fit about 15 playbills snugly. I store them chronologically, and in the back of each sleeve I keep the tickets, slips, and any photos I took with the cast members at stage door. So it has a ton of sentimental value for me now!
I don't know how binders would translate for hundreds?! of playbills. I'm sure you can pack in potentially up to 30 in a binder if you are okay with it looking a bit stuffed.
Photocopy of the cast page is such a great idea!