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There are a few papers by Ahmed Sebbar where he expresses this function in terms of more well known modular forms. (though the expressions are fairly ungainly). There is a relationship with paperfolding and with automatic sequences

On Two Lacunary Series and Modular Curves

Paperfolding and modular functions

This paper came out a few days ago:

Automatic sequences defined by Theta functions and some infinite products by Shuo Li

The series $\sum_{n=1}^{\infty} \frac{1}{2^{2^{n}}}$ is in some contexts known as The Kempner number, and it is known to be transcendental:

In The Many Faces of the Kempner Number, Boris Adamczewski relates a few proofs of its transcendentality.