Closed form of a Fredholm number

It is known that the so called Fredholm number $\sum _{n=0} ^{\infty} \beta^{2^n}$ is transcendental for any algebraic $\beta$ with $0<|\beta|<1$. However, it may be the case, that a closed form expression exists in terms of known transcendental constants ($\pi$, $e$, $\Gamma (\frac 1 4)$, maybe $\gamma$). Does someone know of a way that such a $\beta$ can be construced? Or prove that no such closed form exist?
I am aware that there is little chance for anybody to answer this question, because the existence of a closed form is not known for the much more studied case of $\zeta (3)$.

• What precisely is a "known transcendental constant"? It seems to me that we don't know a whole load more about $\Gamma(1/4)$ or $\gamma$ than we do about $\sum_n 1/2^{2^n}$, so in what sense can you say that the first two are "known" while the third is not? – David Loeffler Apr 6 '18 at 21:17
• @DavidLoeffler That is a good point, and I cosidered it myself when I asked the question. I'd be very happy with using only $e$ and $\pi$, but it seems almost impossible to do so. Therefore, if anyone can find a $\beta$, whose Fredholm number could be expressed in terms of the $\Gamma$-function or $\gamma$, I would accept his answer without hesitation. I think that the point I'm trying to make is some vague notion of the number's "mathematical fame" - e. g. $\gamma$ is much more famous than Catalan's constant $G$ and so on... – FusRoDah Apr 7 '18 at 13:32
• Unless you can define "mathematical fame" and/or "known transcendental constants" precisely, your question is meaningless (which might be why it's attracted several close votes). – David Loeffler Apr 9 '18 at 6:46
• Perhaps you are actually asking for amazing "coincidences" where two numbers with short descriptions happen to agree? – Sam Nead Feb 7 at 8:53