TITLE: Is there a known function defined by the sum of $x^{2^n}$?
QUESTION [7 upvotes]: Is $$\phi(x):=\sum_{n=0}^\infty x^{2^n}$$ a known special function ?

REPLY [2 votes]: 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.