As far as computable ordinals go, the ordinal collapsing function goes pretty far. Here is one, for example:
$$C(\alpha)_0=\{0\}\\C(\alpha)_{n+1}=C(\alpha)_n\cup\{\gamma+\delta,\omega^\delta,\omega_\gamma,\psi_\gamma(\eta)|\gamma,\delta,\eta\in C(\alpha)_n,\eta\in\alpha\}\\C(\alpha)=\bigcup_{n\in\omega}C(\alpha)_n\\\psi_\beta(\alpha)=\sup(C(\alpha)\cap\omega_{\beta+1})$$
It then follows that $\psi_0(\alpha)$ produces extremely large computable ordinals. A couple of values:
$\psi_0(0)=\sup\left\{\omega,\omega^\omega,\omega^{\omega^\omega},\dots\right\}\\ \psi_0(1)=\sup\left\{\psi_0(0),\psi_0(0)^{\psi_0(0)},\psi_0(0)^{\psi_0(0)^{\psi_0(0)}},\dots\right\}$
$\psi_0(2)=\sup\left\{\psi_0(1),\psi_0(1)^{\psi_0(1)},\psi_0(1)^{\psi_0(1)^{\psi_0(1)}},\dots\right\}$
$\vdots$
$\psi_0(\omega)=\sup\left\{\psi_0(1),\psi_0(2),\psi_0(3),\dots\right\}\\ \psi_0(\omega+1)=\sup\left\{\psi_0(\omega),\psi_0(\omega)^{\psi_0(\omega)},\psi_0(\omega)^{\psi_0(\omega)^{\psi_0(\omega)}},\dots\right\} $
$\vdots$
$\psi_0(\zeta_0)=\psi_0(\omega_1)=\sup\{\psi_0(0),\psi_0(\psi_0(0)),\psi_0(\psi_0(\psi_0(0))),\dots\}\\ \psi_0(\omega_1+1)=\sup\left\{\psi_0(\omega_1),\psi_0(\omega_1)^{\psi_0(\omega_1)},\psi_0(\omega_1)^{\psi_0(\omega_1)^{\psi_0(\omega_1)}},\dots\right\}$
And it just keeps going from there. If we were to define an ordinal as follows:
$D(\beta)_0=\{0\}\\D(\beta)=D(\beta)\cup\{\gamma+\delta,\omega_\gamma,\phi(\eta)|\gamma,\delta,\eta\in D(\alpha),\alpha\in\beta,\eta\in\beta\}\\\phi(\beta)=\sup D(\beta)$
The last rule pertaining to limit ordinals. Within the above notation, the computable supremum is given by...
$$\psi_0(\omega_\star)$$
where $\omega_\star$ is the first fixed-point of $\gamma\mapsto\omega_\gamma$ given by $\sup\{\omega,\omega_\omega,\omega_{\omega_\omega},\dots\}$.
And you can write all ordinals less than the above through a combination of addition, base omega exponentiation, initial ordinals, and $\psi$, the four operations in our $C$.
There exists much stronger notations that can extend this pretty far, though you start having to include things like the axiom of inaccessible cardinals and stuff, things beyond ZFC.
Another note is that the ordinal collapsing function uses larger uncountable ordinals (many uncountable ordinals) and collapses them down into smaller uncountable ordinals until it reaches countable ordinals, which then yields a large result.
A stronger one that I've made is as follows:
$${\rm Assume~there~exists~a~weakly~compact~cardinal~}K\\B_0(\alpha,\beta)=C_0(\alpha,\beta)=\beta\cup\{0,1,K\}\\B_{n+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\Psi_\eta(\gamma)~|~\gamma,\delta,\eta\in B_n(\alpha,\beta)\land\eta\in\alpha\}\\C_{n+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\psi_\delta^\gamma(\eta)~|~\gamma,\delta,\eta\in C_n(\alpha,\beta)\land\eta\in\alpha\}\\B(\alpha,\beta)=\bigcup_{n\in\omega}B_n(\alpha,\beta)\\C(\alpha,\beta)=\bigcup_{n\in\omega}C_n(\alpha,\beta)\\\Xi[\alpha]=\bigg\{\beta~|~\beta\notin B(\alpha,\beta)\land\alpha\in B(\alpha,\beta)\land\bigcap_{\eta\in\alpha\cap B(\alpha,\beta)}\Xi[\eta]\text{ is stationary}\bigg\}\\\Psi_\alpha=\operatorname{enum}(\Xi[\alpha])\\\psi_\mu^\pi(\alpha)=\min\{\beta\in\Xi[\mu]~|~\beta=\Psi_{\mu+1}(\pi)\cap C(\alpha,\beta)\}$$
This has a limit of $\psi_0^0(\sup(C(0,0)))=\psi_0^0(\sup\{K,\psi_K^K(0),\psi_{\psi_K^K(0)}^K(0),\dots\})$.
Stronger notations may be found on googology.wikia, mainly Deedlit's OCFs and Tarnovski's C. See here.
9/23/19:
Simply because I feel like leaving this here, one can see here for my more recent ordinal collapsing which uses Greatly Mahlos (and generalizations) instead of the weakly compacts like above.