I want to define a "$t^\zeta$-unreachable" where $t^\zeta$ stands for a tuple of size $\zeta$. I'd call that tuple as the index of reachability of the cardinal in question.
I'll start with $0$-unreachability which actually I take it here to mean accessibility. That is a cardinal $\kappa$ is $0$-unreachable if and only if it is empty or it is subnumerous to the power set of the union of a set $X$ of cardinals smaller than $\kappa$, where $|X| < \kappa$. (where subnumerous means smaller than or equal in cardinality).
Now by $1$-unreachable, I mean a cardinal $\kappa$ that is not $0$-unreachable and such that only the $0$-unreachables less than it are unbounded in it, it's obvious here that every $1$-unreachable cardinal is $\aleph_0$ or an inaccessible cardinal.
Similarly every $\alpha$-unreachable is not $\beta$-unreachable for all $\beta < \alpha$, and for each $\beta$ the set of all $\beta$-unreachables smaller than it is unbounded in it if and only if $\beta < \alpha$
Now we reach into a cardinal $\kappa$ that is $\kappa$-unreachable! Here this will be the point of change, this will be expressed by having a tuple with two entries in it, more precisely as: $[0,1]$-unreachable. (this would be a hyper-inaccessible cardinal).
We repeat the same process above, so an $[\alpha, \zeta]$-unreachable is not a $[\beta, \zeta]$-unreachable for all $\beta < \alpha$, and for each $\beta$ the set of all $[\beta, \zeta]$-unreachables smaller than it is unbounded in it if and only if $\beta < \alpha$
Now we reach into a cardinal $\kappa$ that is $[\kappa, \zeta]$-unreachable, now this would be expressed as $[0,\zeta+1]$, and so on...
We run the above process till we reach into a cardinal $\kappa$ that is $[0,\kappa]$-unreachable, now at that stage this would be expressed by a ternary tuple as: $[0,0,1]$-unreachable!
Now we continue running this process until we reach into a cardinal $\lambda$ that is $t^\lambda$-unreachable, the first one would be of the form "$[0,0,0,....,1(\text{ at } \lambda^{th} \text { entry })]$-unreachable".
The question is:
What large cardinal property $\lambda$ is equal to?
From the Wikipedia page on inaccessible cardinal, I got the impression that a hyper-inaccessible is just an $[\alpha, 1]$-unreachable, while a hyper-hyper-inaccessible (i.e. a hyper$^2$-inaccessible) is an $[\alpha, 2]$-unreachable, and generally a hyper$^\delta$-inaccessible is just an $[\alpha, \delta]$-unreachable. So an $[\alpha,\beta,\gamma]$-unreachable can be a Mahlo. However, I might be wrong since I don't have that parcel of knowledge about large cardinal properties.
Note: I've changed the naming from inaccessible to unreachable in order to avoid confusion with a known terminology as advised in the comments.