Why, intuitively, does the Ackermann function require $\mu$-minimisation?
Roughly speaking, a function is primitive recursive if it can be computed by a program in which all loops are bounded ahead of time. Concretely, we can consider the LOOP programming language, which has the following instructions:

Set $x$ to zero.
Increment $x$.
Run a piece of code $x$ times.

We can use LOOP to compute one-argument functions using some input/output convention, say the input is in $x$ and the output is in $y$.
If the program has no loops, then $y \leq x + C$ for some constant $C$. If the program has no nested loops, then $y = O(x)$. If it has nesting depth 2 (a loop can appear within another loop, but that's it), then $y = O(x^C)$ for some constant $C$. More generally, if the program has nesting depth $d$, then $y$ can be bounded by a function on level $d$ on the Ackermann hierarchy.
The Ackermann function is constructed in such a way that it doesn't belong to the Ackermann hierarchy, and so cannot be computed by a LOOP program. In brief, it grows too fast.

Growing too fast isn't the only reason for not being primitive recursive. Indeed, using diagonalization we can construct Boolean recursive functions which are not primitive recursive. Let $f_i$ be an effective enumeration of all primitive recursive functions, and define
$$
f(n) = \begin{cases}
1 &amp; \text{if } f_n(n) = 0, \\
0 &amp; \text{otherwise}.
\end{cases}
$$
The function $f$ is recursive but, by construction, not primitive recursive.