Is there purely recursive functions?
The Böhm–Jacopini theorem states that everything can be implemented using just IFs and WHILEs. Indeed, your CPU essentially implements an infinite WHILE loop with conditional execution.

Recursion can be simulated iteratively using a stack; this is how recursion is implemented under the hood. Conversely, the same construction as in the theorem shows that everything can be implemented using only tail recursion (which is essentially a WHILE loop). So iteration and recursion are equivalent.