Mahlo cardinal and hyper k-inaccessible cardinal It is known that every Mahlo cardinal $\kappa$ is hyper $\kappa$-inaccessible. It the converse true, namely: every cadinal $\kappa$ which is  hyper $\kappa$-inaccessible is a Mahlo cardinal ?
 A: The answer is no. Mahloness is much stronger than this. 
Every Mahlo cardinal $\kappa$ is a limit of such cardinals. One can see this, because there is a club of $\gamma<\kappa$ with $V_\gamma\prec V_\kappa$, and by Mahloness, we can find such a $\gamma$ that is inaccessible. Since the degrees of hyper-inaccessibility for $\kappa$ are expressible in the structure $\langle V_\kappa,\in\rangle$ (plus the inaccessibility of $\kappa$), it follows that $\gamma$ will inherit all the same hyper-inaccessibility properties that $\kappa$ has. 
The Mahloness of $\kappa$ is fundamentally a second-order property about $V_\kappa$, even given inaccessibility, for it cannot be expressed by inaccessibility plus any first order statement in $\langle V_\kappa,\in\rangle$. One can also see this directly by forcing to add a club $C\subset\kappa$ disjoint from the regular cardinals, forcing with conditions that are closed bounded sets $c\subset\kappa$ containing no regular cardinals, ordered by end-extension. This forcing is $<\kappa$-distributive and hence adds no bounded sets to $\kappa$. So it preserves $V_\kappa$ and the inaccessibility of $\kappa$, but it kills the Mahloness of $\kappa$. So in the forcing extension $V[C]$, the cardinal $\kappa$ remains inaccessible and has exactly the same $V_\kappa$ as before, but is no longer Mahlo. 
Erin Carmody wrote her dissertation, Forcing to change large cardinal strength, following a theme generalizing this idea, and you may be interested in her account of the hyper degrees of inaccessibility and Mahloness, which goes through the richly inaccessible, utterly inaccessible, deeply inaccessible, vastly and so on, introducing an ordinal notation system that unifies these concepts. For example, one has the $(\Omega^{\omega^3+5}\cdot\omega^{17}+\Omega\cdot 100+7)-$inaccessible cardinals, and so on.
A: A nice alternative definition:
$\kappa$ is Mahlo iff every normal function $f:\kappa\mapsto\kappa$ has a regular fixed-point.
This is a direct rewrite of the normal definition. Note that $f$ denotes a club set and the fixed-point is where it intersects a regular ordinal.
One can then show more generally that if every normal $\kappa\mapsto\kappa$ function has a fixed-point, then $\kappa$ is regular.
Clearly $\kappa$ must also be a limit of regulars, since we can make functions with $f(\alpha)>\beta$ for each $\beta<\kappa$.
By considering $f(\alpha)=\aleph_\alpha$, we can also see that a Mahlo cardinal must be greater than the least inaccessible. In general, by considering cardinals greater than the $\beta$th inaccessible, one can see a Mahlo cardinal must also be a limit of inaccessibles.
And so on, one can continue this sort of argument to show that Mahlo cardinals are much greater than any inaccessible type of generalization.
