TITLE: Infinite series where each term is the square of the last
QUESTION [8 upvotes]: Is there a closed-form, in terms of elementary functions or otherwise, for the power series $x+x^2+x^4+x^8+x^{16}+...$, where each term is the square of the last?

REPLY [5 votes]: The series
$$\sum_{n=0}^\infty x^{\large-2^n}$$
generally does not have a closed form. This is just your series where $x \mapsto \dfrac{1}{x}$.
When $2 \le x \le 10$, the decimal expansion is given by the OEIS.  When $x=2$, the number is called the "Kempner-Mahler number." The case when $x=10$ seems to be called the "Fredholm-Rueppel Sequence" and has many other interesting properties.
It has also been shown that the number, $M$, generated by the sum $x=2$ is transcendental by Mahler, and Knight showed that this was true for all $x\ge 2$. (Summarized here)
The continued fraction for this series is discussed for $x \ge 3$ in J. Shallit's "Simple continued fractions for some irrational numbers."