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I would worry they will lose their crispness in the fridge
Put them in an air tight container. No refrigeration necessary.
Every meringue I've ever made--and I made some a week ago--have last a VERY long time in a room temp airtight container as long as they were properly crisp in the first place. And I agree that refrigeration might make them sticky.
What is the meringue for? Refrigeration will definitely add moisture to your meringue, which you don't want.
If it's for a pavlova and your meringue is crisp on the outside and chewy on the inside, I'd make it no more than a few days in advance. If these are the kind of meringues that are really crispy, though, you could for sure make them a week in advance and store them in an airtight container at room temperature.
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