The Ackermann function is a well-known example of a recursive function that is not primitive recursive. It serves as a benchmark for computing and illustrates the concept of deep recursion.
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To get an intuition for it, see the sample computation at: en.wikipedia.org/w/index.php?title=Ackermann_function&oldid=1170238965#TRS,_based_on_2-ary_function where in this context. From this, we immediately get the intuition that these functions are recursive somehow.