Is $\sum_{k=0}^\infty \frac{1}{k!!} x^{k!}$ non-elementary?

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Is $z \mapsto \sum_{k=0}^\infty \frac{1}{(k!)!} z^{k!}$ an elementary function? I designed it to be analytic but analogous to Liouville's constant, but don't know how to search for this function.