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
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
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.