The function $f(\alpha)$ outputs an uncountable ordinal from an arbitrary ordinal $\alpha$. It is defined as follows:
$$f(\alpha) = \left\{ \begin{array}{l}
{\omega _1}\quad {\text{if}}\quad\alpha = 0{\rm{;}}\\
{\omega _{f(\alpha - 1)}}\quad {\rm{if }}\quad\alpha {\text{ is a successor ordinal;}}\\
{\rm{sup\{ }}f(\beta ):\beta < \alpha {\text{\} }}\quad {\text{if }}\quad\alpha {\text{ is a limit ordinal}}{\rm{.}}
\end{array} \right.$$
Let $\gamma$ denote the cardinal such that $f(\omega_1)$ is the initial ordinal of $\gamma$. How large is $\gamma$ in the hierarchy of large cardinals? For example, how to place $\gamma$ in the hierarchy of Cantor’s Attic?
(If $f(\alpha)$ is not well-defined for any ordinal $\alpha$, I need an explanation.)
Since all of this is defined in $\sf ZFC$, the term "large" is not very applicable here.
The function spits out $\aleph$ fixed points at all limit steps, and indeed the starting value of $\omega_1$ is irrelevant, as long as it is below the first fixed point.
So $f(\omega_1)$ is simply the $\omega_1$th fixed point, which is also the first fixed point whose cofinality is uncountable. It's not a very impressive cardinal, to be honest. It's big, sure, but most cardinals are bigger.
For example, to see that it's not very large, $V_\gamma$ is able to detect that there is a cofinal sequence of ordinals of type $\omega_1$.
If you want to grow faster, you can start skipping a lot more cardinals. For example:
Define $g(0)$ to be the least fixed point; $g(\alpha+1)$ is the $g(\alpha)$th $(g(\alpha))$-fixed point; limits at limit ordinals.
Here $(\alpha)$-fixed point would be defined recursively as well, $(0)$-fixed point is just any cardinal; the class of $(\alpha+1)$-fixed point is the fixed points of the enumerations of $(\alpha)$-fixed points; and for a limit ordinal $\alpha$, an $(\alpha)$-fixed point is a fixed point of all $(\beta)$-fixed points for $\beta<\alpha$.
Now $g(1)$ is so much bigger than your $\gamma$. Still, $g(\omega_1)$ is still a very small cardinal, all things considered.
(Because all of this takes place in $\sf ZFC$.)