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?