From Gerry's answer it follows that the existence of closed form for the required function is reduced to the existence of closed formula for $S_2(n)$, the sum of digits in the binary record of $n$. Note that a reasonable generating function for this series, $$\sum_{n=0}^\infty x^{S_2(n)}q^n =\prod_{n=1}^\infty (1+xq^{2^n})$$ is well studied in transcendence number theory; see, for example, [J.M. Borwein and P.B. Borwein, Amer. Math. Monthly 99 (1992) 622-–640] and links therein.
P.S. Some of you may enjoy classics---an elegant proof by J. Liouville from 1840 of the non-quadraticity of $e^2$. The proof makes a very clever use of the $2$-adic order of $n!$
From Gerry's answer it follows that the existence of closed form for the required function is reduced to the existence of closed formula for $S_2(n)$, the sum of digits in the binary record of $n$. Note that a reasonable generating function for this series, $$\sum_{n=0}^\infty x^{S_2(n)}q^n =\prod_{n=1}^\infty (1+xq^{2^n})$$ is well studied in transcendence number theory; see, for example, [J.M. Borwein and P.B. Borwein, Amer. Math. Monthly 99 (1992) 622-–640] and links therein.