TITLE: What methods show that a number is transcendental?
QUESTION [8 upvotes]: I've been doing a lot of research on such theories lately and these are all I've found so far:
Liouvilles criterion (here)
Lindemann-Weierstrass theorem (here)
Gelfond-Schneider theorem (here)
Brownawell-Waldschmidt theorem (here or here)
Schanuels conjecture (no proof yet) (here)
Six and five exponentials theorem (here)
Bakers theorem (here)
Roths theorem (here)
Or theorems such as:
If you take two transcendental numbers, $a$ and $b$, then at least one of $ab$ and $a+b$ is transcendental (here)
or
$x^{x^x}$ is transcencental if $x\in\mathbb{Q}, x\notin\mathbb{N},x>0$ (here)
What other methods show that a number is transcendental besides these ones?
If you know any other proven or unproven theorems it would be nice to share them. 
Edit 1
K.Dilcher and K.B. Stolarsky (here)
Carl Ludwig Siegel (here)
Edit 2
The Rogers-Ramanujan continued fraction (here) is transcendental for all algebraic $-1<q<1$
All the transcendental functions (here)
Edit 3
All about trancendental Infinite Products (here) (here)
and (here)

REPLY [3 votes]: Here they say that a number $$x=\prod_{n=1}^\infty \frac {⌊\alpha^{a_n}⌋}{\alpha^{a_n}} $$ is transcendental if $$\lim_{n\to\infty}\inf \frac{a_{n+1}}{a_n} > 2$$ and if $\alpha$ is a real algebraic non-integer number such that no power of $\alpha$ is a Pisot number and $a_n$ is a sequence of positive integers.
I think that this result is astonishing