We discussed Why does Man ask Why questions?
Following that, I 'why?' has this feature moving 'up' or 'out' of the layer of explanation. And that from an anti-foundationalist stance, we find these explanations eventually looping, or at least leaning on each other - like say Deutsche's 4 methodoligies in The Fabric Of Reality. Hofstadter finds specifically tangled hierarchies of knowledge, with looping and self-reference, to be characteristic of minds or subjectivities.
So, I would say the recursions are useful up to a point. Everyone asks why. Philosophers ask why we ask why. Meta-philosophy and meta-mathematics go more abstract still.
We can also invert the procedure:
"The whole problem of justifying nature, of trying to make life mean
something in terms of its future, disappears utterly. Obviously, it
all exists for this moment. It is a dance, and when you are dancing
you are not intent on getting somewhere… The meaning and purpose of
dancing is the dance." - Alan Watts
But in considering infinite steps, I would look at another recursion: Who watches the watchmen? We have a structure of scrutiny, within structures, and between them. We have circles of accountability, like say division of powers. And we can go inward, impulse control, superego. But the conceptual 'watches' starts to fray, and transform, as the domain alters. Does your superego 'watch' you? Do judges 'watch', their power is more a backstop, a failsafe. And isn't it, the people, and the social contract, that is the real 'watcher' of the executive?
I'd say around where we ask, why we ask why we ask why we ask why, you have a complex enough hierarchy to start tangling it, creating structures of internal consistency, and the 'weave' of different modes working together, where we judge 'how well things hang together', rather than say foundations and steps. Because the loops can recur infinitely without going anywhere - more than on the horns of Munchausen's trident, instead in our own strange-loop satisfaction with understanding our place in our cosmos.
Edited to add:
I was just reading: In physics, the fourth, fifth and sixth derivatives of position are defined as derivatives of the position vector with respect to time – with the first, second, and third derivatives being velocity, acceleration, and jerk. Although rarely used, the next differentials are snap, crackle, and pop. Spacestations set limits on jerk and snap. I feel there is an analogy to be drawn to the linguistic effect of applying why to why, with kind of taking a derivative.