It follows the discussion of this post:
I know axiom of regularity forbids $x=\{x\}$, (since $x \in x$ is forbidden by axiom of regularity)
Does axiom of regularity allow $x=\{\{x\}\}$? (My try: $\{x\} \cap x = \emptyset$ can be satisfied, it seemed not to contradict axiom of regularity.)
More generally, are more level of nesting like $x=\{\{\{x\}\}\}$,$x=\{\{\{\{x\}\}\}\}$ ... etc allowed?
Apply regularity to $\{\{x\}, x\}$. The possible elements are $\{x\}$ and $x$. We have $\{\{x\}, x\} \cap \{x\} = \{x\} \neq \emptyset$. On the other hand, if $x = \{\{x\}\}$ then $\{\{x\}, x\} \cap x = \{\{x\}, x\} \cap \{\{x\}\} = \{\{x\}\} \neq \emptyset$. Thus regularity implies that $x \neq \{\{x\}\}$.
To show $x \neq \{\{\{x\}\}\}$, apply regularity to $\{\{\{x\}\}, \{x\}, x\}$. You can generalize this to show that $x \neq \{\cdots\{x\}\cdots\}$ for any number of iterated brackets.
