Is there a special word or phrase for a CD with a carefully picked playlist you made for a particular person, often given as a gift?
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As an alternative to 'mix CD,' I've often just called them compilations.
The only problem with calling them simply 'compilations' is that it doesn't really differentiate from a commercially released compilation album, so you would have to say 'my own compilation' or 'compilation that I made.'
So it's not nearly as nice and succinct as simply 'mix CD,' but it still works.