unnamed factorial

im bad at extending this to make it sound "formal" so i will just dump the definition

Notated as: $n?$

$0? = 1$

$n? = n!^{(n-1)?}=n\underbrace{ !!\cdots!! }_{(n-1)?}$ for $n > 0$(n!! . . !! as on iteration, not as on the double factorial)

First few values

1? = 1

2? = 2

3? = 3!! = 6! = 720

4? = 4!!! . . (720 !'s) . . !!! $\approx$538 PT (1.449268888905 × 10²⁵) $\approx$(not so good approximation) E25#539

5? = no idea how to calculate

any sugestions to how i shall name this? i have considered interrogatorial (or ?torial) and detonatorial (detonate + factorial, kinda obvious), i also thought of making it the fourth superfactorial, since it beats all of them

edit: i dont think i wanna name it detonatorial, so either i name interrogatorial (or ?torial) or i make a superfactorial