Most "simple" $\mu$-recursive function that is not primitive recursive Maybe the most prominent example of a $\mu$-recursive function that is not primitive recursive is the Ackermann function. But writing it out as a $\mu$-recursive function ("breaking it all the way down to the elementary operators") is rather complicated (see here).
I wonder what a most "simple" example of a $\mu$-recursive function that is not primitive recursive would be (without trying to define what exactly "simple" means). Or is the Ackermann function maybe maximally simple?
 A: I know this is probably not what you wanted but the simplest function that comes to my mind is :
Let $f : x\mapsto 1$, the constant function equals to $1$.
Then let $g : y\mapsto \mu_xf(x)$. 
$g$ is a function that never halts. $g$ is very simple and not primitive recursive.
A: The definition of the Ackermann function is simple, since it only takes a few lines.
So, I presume you intend "simple" in a different sense, possibly in relation
to the computational complexity of the function or the logical difficulty of
proving its totality.
From the complexity point of view, there is an important result concerning many 
interesting complexity classes (belonging to the "folklore" of the subject), 
that is the following downward closure property: if a function f is computable 
in time O(g) and
g is in a class C, than f is in C too (provided C is, say, closed w.r.t 
polynomials).
The result is a more or less trivial consequence of the complexity of Kleene's
T-predicate (i.e. bound interpretation) for Turing machines, that is very low: 
not only it is primitive recursive: it is polynomial too. 
So, if you know the upper bound g in time, you can run a bound interpreter for
a suitable machine for the expected time g and hence simulate an arbitrary 
computation with a minimal overhead w.r.t. g.
This means that to look for a function that is not primitive recursive you need
to look for a function whose computational complexity is not primitive 
recursive, and the simplest way to define such a function is to look for a 
function whose output grows sufficiently fast (since in order to produce
an output n, you need at least time n). 
From  the logical point of view, the simplicity of the Ackermann function 
is manifested by the fact that it is easily definable in system T: 
by the well know 
characterization of Godel, this means that its totality can be already 
proved in Peano arithmetics, that is a relatively weak logical system.
Another way to understand how close is Ackermann function to the primitive 
recursive setting is the following. Suppose to relax a bit the condition 
of primitive
recursion, allowing recursive calls where arguments are smaller in
lexicographic order with respect to the arguments of the caller.
Intuitively, this is already enough to ensure the well foundedness of
the recursion, and hence the mathematical soundness of the definition.
With this slightly relaxed definition of "acceptable recursion", Ackermann
function would become a perfectly legal definition.
