The Ackermann function is a well-known example of a recursive function that is not primitive recursive. It was developed by Wilhelm Ackermann in the early 20th century to illustrate the concept of computability and the limits of recursive functions. The function takes two non-negative integer inputs and produces a single non-negative integer output, growing extremely fast as the inputs increase. The Ackermann function is defined using a set of rules that involve multiple cases, making it a complex yet fascinating example in theoretical computer science. It serves as a benchmark for evaluating the efficiency of algorithms and the power of different computational models, highlighting the differences between simple recursion and more complex forms of computation.