Strictly speaking, it isn't a universal phenomenon. For example, stationary limits do not arise in the hyperinaccessibility hierarchy between inaccessible and Mahlo cardinals, nor in the hierarchy of worldy cardinals nor in hyperworldy nor hyper-otherwordly.

A cardinal is $\alpha$-inaccessible if it is inaccessible and for every $\beta<\alpha$ it is a limit of $\beta$-inaccessible cardinals. Thus, the $0$-inaccessible cardinals are exactly the inaccessible cardinals, and the $1$-inaccessibles are the inaccessible limits of inaccessibles, and the $2$-inaccessibles are the inaccessible limits of inaccessible limits of inaccessible cardinals, and so on.

Meanwhile, for $\kappa$ to be *hyperinaccessible* means that $\kappa$ is $\kappa$-inaccessible, and then one defines $\alpha$-hyperinaccessible and so on. One proceeds to the hyper-hyper-inaccessibles and the hyper${}^\alpha$-inaccessible and eventually to the *richly* inaccessible cardinals, and the *utterly* inaccessible cardinals and the *deeply* inaccessible cardinals, and so on.

For $\kappa$ to be a stationary limit of inaccessible cardinals would imply that $\kappa$ is hyperinaccessible and richly so, and deeply so, and utterly so, and so on, already swamping this entire hierarchy. Thus, the hierarchy of inaccessibility is a little more refined and slower than the hierarchy requiring stationary limits and stationary reflection.

A similar distinction plays out lower down with the worldy cardinals, and the hierarchy of hyperworldliness and otherwordliness and hyperotherwordliness. To insist upon stationary limits here would be to jump immediately up to Mahloness.

In summary, my answer is that stationary limits are a convenient way of jumping higher in the large cardinal hierarchy, but there are also many well-studied ways of jumping not quite so high, and there is a rich hierarchy of lesser notions falling short of stationary limits.

in measure, for some measure. This is vastly stronger, of course. $\endgroup$