Convince me I'm wrong!!

Okay, Internets:

I have a dilemma. Well, not a dilemma so much as a need for answers. Here's the Problem:

My Daughter, being our first and only child, has managed to accumulate quite the menagerie of stuffed animals. Naturally, she is only really aware that about five of them exist, but we have at least 20-30 in rotation through our house at the time, and about 20-30 in a Rubbermaid bin in the basement. So...

In my ideal world, she would pick the five she couldn't live without and we would give the rest away to charity. But, of course, anytime she is asked to identify the ones to give away to children who don't have ANY stuffed animals, she wants to keep them all, and then when I find them all a "place" to live in the room, she sees them all lined up nice and neat and immediately has to play with them. All. So the stuffed animals end up two days later back in the regular spots- between the bed and the wall, behind/under various pieces of furniture, or piled in a pile and forgotten for a a few months until we start the dance all over again.

Now, I have never understood the importance of stuffed animals. I never slept with them and I never took them everywhere I went. By the time I hit middle school I had two stuffed animals which were kept more for my mother's nostalgia than for mine. I told all my boyfriends that they were never to give stuffed animals at any approaching gift-giving occasions. So now I feel equally ambiguous toward all but maybe two of the stuffed animals my daughter owns, and I would really like to not spend any more time shuffling them from place to place. I also should note that The Girl has never become especially attached to any one of them, nor has she ever even noticed whether or not we take one when we travel.

So, do I just do the surreptitious mommy-cleaning-while-kid's-away routine and make them all disappear one by one? Or should I find a way for them to live with us until the girl is old enough to move them out by her own decision and/or manpower? And most importantly, if I'm keeping them... HOW?? What are your best ideas for stuffed-animal storage that keeps them out of the floor but accessible/visible for playing??

I need your help, Internets. You're my only hope!!