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At nursery my dd has a toothbrush, 2 changes of clothes, hat (sun/winter as appropriate), suncream and slippers. I send her in with waterproofs and wellies if it is wet as I am to stingy to buy spares. When she was having a nap they also kept a dummy there.
Our nursery provides disposable nappies and pull ups. When dd was in nappies I sent in a second bag with 4 washable nappies in, they kept it in the bathroom to make changing easier. I didn't bother with the washable wipes at nursery, thought it would be too much hassle for them.