A full-size Mickey straw will work (I haven't bought one in a while but you used to get them right at your resort's food court, usually around the check-out). I specify size because I know I've seen smaller Mickey straws around property too (like the ones you get with those decorated marshmallow treats). Anyway, yes, the big straws will work with a refillable mug, and if they don't go exactly to the bottom, it's very close.
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