Logic of the even weaker excluded middle

Question

There exists a superintuitionistic propositional logic where $\neg p \vee \neg \neg p$ is a theorem, but $p \vee \neg p$ is not a theorem. It is called the logic of the weak excluded middle. That raises the question, is there a superintuitionistic propositional logic where $\neg \neg p \vee \neg \neg \neg p$ is a theorem, but $\neg p \vee \neg \neg p$ is not a theorem? Of course, the question can be generalized by adding more negation signs.

Answer 1

No - perhaps surprisingly, intuitionistic logic does prove triple negation elimination $$\neg\neg\neg p\equiv \neg p.$$ So this collapses the "LEM-hierarchy" right at the second level.