Is there a known function defined by the sum of $x^{2^n}$? Is $$\phi(x):=\sum_{n=0}^\infty x^{2^n}$$ a known special function ?
 A: Not an answer but just a note on $x\to 1$ (following [1] and [2] closely).
Basically, $\phi(x)=-\log_2(1-x)+\mathcal{O}(1)$ with the "$\mathcal{O}(1)$" oscillating:
$$\psi(x)=\phi(x)+\frac{\ln(-\ln x)}{\ln 2}+\sum_{n=1}^{\infty}\frac{(\ln x)^n}{n!(2^n-1)}$$
satisfies $\psi(x)=\psi(x^2)$. The "Mellin transform approach" gives, in our case,
$$\psi\big(e^{-2^{-x}}\big)=\frac{1}{2}-\frac{\gamma}{\ln 2}+\frac{1}{\ln 2}\sum_{n\in\mathbb{Z}\setminus\{0\}}\Gamma\Big(\frac{2n\pi i}{\ln 2}\Big)e^{2nx\pi i}.$$
A: 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.
