Is there a "natural" example of a total, computable but non-primitive recursive function?
This answer gives us a hint, namely that an interpreter for a language more powerful than primitive recursion, is itself not primitive recursive. For example, System F has no self-interpreter, and thus any interpreter for System F is not self-recursive. 

This then extends to Turing-complete languages: JavaScript, Python, Ruby, etc. Any interpreter you run is utilizing functions more powerful than primitive recursion.