Q:

Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related to these functions, but I am not sure if they still keep the property of being NOT primitive recursive. My intuition is that they are, but I am not sure. Any textbook or references to read to get a better understanding of this will be very appreciated (and a straight response too!). 
NOTE: I read this related question, but it doesnt help me.

A:

If I have understood well yes. 
Because if follow from the definiton of hyperoperators definition.
$H_0(a,b):=S(b)$
is the successor function, and every hyperoperation is defined recursively from the successor function:
(recursive definition of $H_{n+1}$ using $H_n$)
$i)$ $H_{n+1}(a,0):=a_{n+1}$
$ii)$ $H_{n+1}(a,b+1):=H_n(a,H_{n+1}(a,b))$
Here $a_{n+1}$ is the initial value of the function when the argument is $0$ and in the case of hyporoperators we have that 
$a_ {n+1}:=
\begin{cases}
a,  & \text{if $n=0$} \\
0, & \text{if $n=1$ }  \\
1, & \text{if $n\gt 1$ }  \\
\end{cases}$
Since $H_0(a,b)=S(b)$ is the successor funtion and is basic primitive rucursive funtion, we have that all the hyperoperations (that are defined from $H_0(a,b)=S(b)$) are Primitve recursive functions.

A:

(Elaborating on my comment to the accepted answer ...) 
For each $n \in \mathbb{N}$, the $n$th hyperoperation 
$$H_n: \mathbb{N}^2 \to \mathbb{N}: (x,y) \mapsto H_n(x,y)$$ 
is primitive recursive.
However, for any integer constant $c\ge 2$, the following functions are not primitive recursive:
$$\begin{align}
f_c&: \mathbb{N} \to \mathbb{N}: n \mapsto H_n(c,n)\\
g_c&: \mathbb{N}^2 \to \mathbb{N}: (n,y) \mapsto H_n(c,y)\\
h&: \mathbb{N}^3 \to \mathbb{N}: (n,x,y) \mapsto H_n(x,y).
\end{align}$$
When $n$ is included as an argument, the function grows too fast to be primitive recursive.
E.g., $x\uparrow^n y\ $ is primitive recursive as a function of $(x,y)$ for fixed $n$, but it is not primitive recursive as a function of $(n,y)$ for fixed $x\ge 2$. 
Essentially these results are proved in Chapter 13 of  "Computability, Complexity, and Languages" (1983) by Davis & Weyuker, and also sketched in Introduction to Computability Theory by Zucker & Pretorius.

