Oddly enough, this is the structure of my own children's book, How the Fisherman Tricked the Genie. It's based fairly closely on a lesser known story from the Arabian Nights.
The outer story is about a fisherman who has the misfortune to release an evil genie from a container. The fisherman tells the genie a fairy tale about a wicked king and a magician, who in turn tells a fable about a prince and a dog. In my version there's a more pronounced difference of styles between the stories, but the nested structure is present in the original. Arguably, the entire Arabian Nights is itself an example of this structure. Although it's not exactly a "folktale" it almost certainly compiles stories from the oral tradition.
I feel certain there must be others, but I'm having trouble thinking of one. The widespread folktale "The Hidden Treasure" (which inspired Paulo Coelho's bestseller The Alchemist) has characters that retell their dreams. Possibly also of interest is the well-known spiritual "Mary, Don't You Weep," which centers around figures from the New Testament, but interpolates Old Testament Bible stories. If you don't require English, there are West African poems that are essentially compilations of allusions to well-known proverbs, fables and older poems. This Wikipedia article, although not focused on the oral tradition, does give some other good examples, including the Odyssey.