Since there aren't many additive inaccessible OCFs, I will attempt to make one. I don't know much about inaccessible OCFs so this might be horribly broken.
Define \(\Omega_\nu\) as follows:
- \(\Omega_0=1\)
- \(\Omega_{\nu+1}=\text{next regular after }\Omega_\nu\)
- \(\Omega_{\nu}=\lim\{\Omega_\mu\}\) where \(\mu>\nu\) if \(\nu\) is a limit ordinal
Define \(I_\alpha\) as the \(\alpha\)th weakly inaccessible.
Define \(\mho_\nu\) as follows:
- \(O_\nu^0=1\)
- \(O_\nu^{n+1}=\{\alpha|\alpha<\Omega_\beta\wedge\beta\in O_\nu^n\wedge\beta<\nu\}\)
- \(O_\nu=\bigcup_{n\in\omega}O_\nu^n\)
- \(\mho_\nu=\min\{\alpha|\alpha>\beta\wedge\beta\in O_\nu\}\)
Define \(\chi\) as follows:
- \(X_\nu^0(\alpha)=I_\nu\)
- \(X_\nu^{n+1}(\alpha)=\{\beta+\gamma,\chi_\nu(\eta)|\beta,\gamma,\eta\in C_\nu^n(\alpha)\wedge\eta<\alpha\}\)
- \(X_\nu(\alpha)=\bigcup_{n\in\omega}X_\nu^n\)
- \(\chi_\nu(\alpha)=\min\{\beta|\beta\notin X_\nu(\alpha)\}\)
Define \(\psi\) as follows:
- \(C_\nu^0(\alpha)=\mho_\nu\)
- \(C_\nu^{n+1}(\alpha)=\{\beta+\gamma,\psi_\mu(\eta),\chi_\iota(\kappa)|\mu,\beta,\gamma,\eta,\theta,\iota,\kappa\in C_\nu^n(\alpha)\wedge\eta<\alpha\}\)
- \(C_\nu(\alpha)=\bigcup_{n<\omega}C_\nu^n(\alpha)\)
- \(\psi_\nu(\alpha)=\min\{\gamma|\gamma\notin C_\nu(\alpha)\}\)