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O(2^2^n) = O(4^n) = O(c^n) ; O(2^3^n) = O(8^n) = O(c^n) ; ...

I think grandparent meant 2^(2^n), not (2^2)^n.

Even so, O(4^n) != O(8^n); there is no constant k such that k * 4^n >= 8^n.