Apologies in advance if this question is obvious/not research level.
Let $\preceq$ be the consistency strength relationship on theories. Working over $ZF$ or $ZFC$, is there some large cardinal notion $\psi(\alpha)$ with an ordinal parameter such that there exists some ordinal $\kappa$ such that $ZF(C)+\psi(\alpha)$ is consistent relative to some stronger theory for all $\alpha<\kappa$ and $$ZF(C)+\psi(\alpha)\prec ZF(C)+\psi(\alpha+1)$$ for all $\alpha<\bigcup\kappa$, but $ZF(C)+\psi(\kappa)$ is inconsistent? What about a $\psi'(\alpha)$ such that $ZF(C)+\psi'(\alpha)$ is consistent relative to some stronger theory and $$ZF(C)+\psi'(\alpha)\prec ZF(C)+\psi'(\alpha+1)$$ for all set-sized $\alpha$, but $ZF(C)+\psi'(O_n)$ is inconsistent?
The motivation is having a large cardinal notion that gets 'as close to inconsistency' as possible without becoming inconsistent, in a precise sense.