“Nested independence” of $\mathsf{ZFC}$? [duplicate]

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• Can unprovability unprovable? Is there an $\omega$-fold unprovability? 4 answers

I have a question about undecidable statements in $\mathsf{ZFC}$.

I know there are true statements like: $$X\text{ is independent of }\mathsf{ZFC}.$$

But is it also possible that such a statement exists?

$$\text{"}X\text{ is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}$$

Or even worse:

$$\text{""}X\text{ is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}\text{" is independent of }\mathsf{ZFC}$$

I think you see what I mean by "nested independence of $\mathsf{ZFC}$". Is it possible that such a true statement exists? If it is possible: What does it mean? And is there an example for such a true statement? Unless such a true statement can exist: Why not?

Another funny case would be is the depth of this nested statement goes to infinity:

$$\begin {align} \mathrm{statement}_0 &= X \\ \mathrm{statement}_{n+1} &= \mathrm{statement}_n\text{ is independent of }\mathsf{ZFC} \\ \mathrm{specialstatement} &= \lim_{n\to\infty}\mathrm{statement}_n \end{align}$$

Does such a $\mathrm{specialstatement}$ exist? And what is the meaning of it?