They have paper cups at the food courts if you just want water. I don't think they make you buy a mug. For the mugs that are "chipped", I think I heard somewhere where they let you refill them about every 15-20 minutes or so. The chip kind of times it, so if someone buys one mug and has 10 people who bring their own cup to the food court, the mug person can't stand there filling up 10 cups all at once.
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I'm a little crazy with giving the kids with me water while in Disney - all 3 of my kids were hospitalized with dehydration as babies and I'm very vigilant about making the kids drink water. I will give them some of the soda while at the restaurant but leave with all mugs full of water to continue them drinking in the room. I buy water bottles and freeze them to take into the parks. Soda does not hydrate you properly too much sugar.