Every park seems to have these humongous things, usually on nearly every game (especially races or ring tosses). The way to win them is to keep playing and winning, because you have to trade up - something like 3 small prizes makes a medium, 3 medium for a large, 3 large for an x-large, 3 x-large for a jumbo. Sheesh! Is it really worth it?
As for all day, I've seen people try to cram them in lockers, and some parks have big lockers/barrels specifically for large items. They've also taken them on some rides, or leave them on benches or with a non-rider while they go on rides.
One year my father threw a hula-hoop over a wooden block at CP, and we won one of those jumbos - some odd mouse with a hat. I had to sit with the stupid thing on the way home, and it sat in my room for a few months before we got rid of it. Whoopee!
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Po!nt of View: A different look at Roller Coasters.
http://www.crosswinds.net/~justmayntz/thrills/index.html