Apologies in advance if this question is obvious/not research level.



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> 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?
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The motivation is having a large cardinal notion that gets 'as close to inconsistency' as possible without becoming inconsistent, in a precise sense.



Cofinal inconsistency
The answer is yes.


One can make easy artificial examples. For example, let $\psi(\gamma)$ assert that there are (only) finitely many inaccessible cardinals, but at least $\gamma$ many.


From the assumption of infinitely many inaccessible cardinals, we can prove the consistency of ZFC+$\psi(n)$ for any particular finite $n$, and furthermore the consistency strength of $\psi(n)$ is steadily increasing as $n$ increases, since in fact $\psi(n+1)$ implies $\text{Con}(\text{ZFC}+\psi(n))$. But $\psi(\omega)$ is clearly inconsistent, so this is an instance of your requested phenomenon with $\kappa=\omega$.


One can make many other similar such examples using larger $\kappa$, simply by taking $\psi(\gamma)$ to assert that there are fewer than $\kappa$ many LC of a certain type, but at least $\gamma$ many.


