You won't have any trouble filling your box. I don't use straw as my lad is very wet, and I don't deep litter, have to clean out every day.....have to!
I had more of a problem a few years ago when I moved to a yard who didn't want shavings on the yard. They encouraged you to uase shredded newspaper, but my old mare used to get tangled in it so I moved to Aubiose (wasn't much choice in those days!)
Oh but, trying not to take all the wet bedding out.....nearly had a breakdown!