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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)

Kinheadpump
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2 Answers2

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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

blausup
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Perhaps start with a book, like

Baker, Alan, Transcendental number theory, London: Cambridge University Press. x, 147 p. (1975). ZBL0297.10013.

GEdgar
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