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In parts of the UK and Commonwealth countries, mixed numbers (aka mixed fractions) are commonly used till about 16 years old. So, mark schemes typically specify $5\frac14$ instead of $\frac{21}4.$ For example, do a keyword search for “mixed”:     https://www....


In addition to "complex fraction" and "compound fraction", they are also sometimes called "nested fractions", as can be shown (for example) here.


I like your term. The wikipedia article on fractions also mentions they are called complex fractions or compound fractions. Personally, I dislike the term complex fraction as it is obviously going to be interpreted as things like $\frac{3+i}{2-i}$. I think I call them "fractions of fractions" which is really in tune with your term composite or the ...


The term complex rational expression (or complex fraction) is commonly used, in U.S. algebra/college algebra texts, to refer to rational expressions where the numerator and/or denominator contain sums or differences of other rational expressions. Note that technically this would not apply to an expression $\frac{a/b}{c/d}$, which has no sums or differences ...


I googled fraction over fraction and was led to the term "complex fraction". Note that this includes more different cases than the one you identified, but does include fraction over fraction. Lots of hits on that term and it is commonly used, for example at Khan Academy. See:

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