A (1-)fibre product (a.k.a. pullback) in a category consists of three distinct objects:
$$\require{AMScd} \begin{CD} X \times_Z Y @>q>> Y \\ @VpVV @VVV \\ X @>>> Z \end{CD}$$
$(X \times_{Z}Y,p,q)$.
Now, in particular we have (often) non-trivial fibre products over the same objects, i.e. $X=Y$, this particular fibre products can be indicated as $X \times_Z X \rightrightarrows X$.
If I want to repeat the operation of fibre product considering $X \times_Z X \times_Z X$ how many arrows associated with this fibre product will I have?I have to take count of the two arrows within the first fibre product, so I think that the question is not trivial.
I'd leave this as a comment, but I want to draw a couple of diagrams...
Suppose you have three arrows $f,g,h\colon X\to Z$ (possibly $f=g=h$, if you are interested in a "fibered cube" of the same object over $Z$). Then you may consider first the pullback with respect to $f$ and $g$:
$$\require{AMScd}\begin{CD} X\times_Z X @>\overline{f}>> X\\ @VV\overline{g}V @VVgV\\ X @>f>> Z \end{CD}$$
And then consider the pullback
$$\require{AMScd}\begin{CD} (X\times_Z X) \times_Z X @>k>> X\\ @VV\overline{h}V @VVhV\\ X\times_Z X @>g\overline{f} = f\overline{g}>> Z \end{CD}$$
This gives you three arrows:
\begin{align*} (X\times_Z X)\times_Z X & \xrightarrow{k} X,\\ (X\times_Z X)\times_Z X & \xrightarrow{\overline{h}} X\times_Z X \xrightarrow{\overline{f}} X,\\ (X\times_Z X)\times_Z X & \xrightarrow{\overline{h}} X\times_Z X \xrightarrow{\overline{g}} X. \end{align*}
In general, $\underbrace{X\times_Z \cdots \times_Z X}_n$ should come with $n$ arrows to $X$.
Is that what you need?