Preliminaries
Let $m,n \in \mathbb{Z}^2_{>0}$ with $m<n$ and $\alpha_1, \alpha_2, \dots, \alpha_n \in \overline{\mathbb{Q}} \setminus \{0,1\}$, with $\alpha_2 \not\in \mathbb{Q}$. We let $\log^n(x)$ denote the $n$-fold composition of $\log$, e.g., $\log^2(x) = \log(\log(x))$. We let ${}^n x$ denote the $n$-fold tetration, e.g., ${}^3 x = x^{x^x}$. We let $\boxed{f}$ denote the set of complex roots of some function $f$. Assume $\alpha \in \overline{\mathbb{Q}} \setminus \mathbb{Q}$, $\beta^\gamma \ne 1$, and $\beta^{\gamma \alpha} \in \overline{\mathbb{Q}}$.
Generally, the expressions discussed below are sets in disguise. For example,
$$
2^{\sqrt{2}} = \boxed{x^{\sqrt{2}/2} - 2} = \left\{ 2^{\sqrt{2}} e^{k \cdot 2\pi i \sqrt{2}} : k \in \mathbb{Z} \right\}
$$
Each element in this set is transcendental by Gelfond–Schneider theorem. $\exp$ and $\log$ are multivalued functions.
We will use SC-Transcendental for numbers which are demonstrably transcendental after we assume Schanuel's conjecture.
Big Set of Transcendentals
The following are SC-Transcendental.
\begin{align*}
&\log^n(\pi),\quad {}^n e,\quad {}^n \pi, {}^{n} \alpha \quad e+\pi,\quad e\pi,\quad e^{\pi^2},\quad \boxed{2^t + 3^t - 1},\quad \sqrt{2}^{\sqrt{3}^{\sqrt{5}}},\\
& e^{e^{\alpha_1}},\quad \sin(e),\quad \cos(e),\quad \tan(e),\quad \sinh(e),\quad \cosh(e),\quad \tanh(e),\\
&\pi^e,\quad \pi^i,\quad 2^\pi,\quad 2^e,\quad \log \pi,\quad \log \log 2,\quad (\log 2)^{\log 3},\quad \sqrt{2}^{\sqrt{2}^{\sqrt{2}}},\\
&i^{i^i},\quad i^{e^{\pi}},\quad \alpha^{\beta^\gamma},\quad e^{e^{\dots^{e^{\alpha_1}}}}, \log(\Gamma(m/n)) + \log(\Gamma(1-m/n)), \boxed{\alpha^{\alpha^x}=x}, \\ & \pi^{\pi^2}, \pi \log(2), 2^{\log(2)}, \log(2)\log(3), \pi 2^\sqrt{2}
\end{align*}
The Gist
We aim to explore the type of intuition Schanuel’s Conjecture provides—specifically, what kinds of transcendence results it allows us to access. We have ${\alpha_1}^{\alpha_2}$ transcendental by Gelfond-Schneider but now we get one more tier as $\alpha^{\beta^\gamma}$ is SC-transcendental. We gain more influence over numbers directly related to $e$. SC enhances our reach into transcendental numbers built from $e$, and through Euler’s formula, into expressions involving $\pi$. We still don't get complete control: $$e^{\pi^{e^{\pi^{\pi}}}}, \sin(\cos(\tanh(e))),
\sum_{n=1}^\infty n^{-3},
\sum_{n=1}^\infty \frac{1}{2^n-1}$$ seem to be non-examples. Not that these could someday be shown to be SC-transcendental (seems unlikely)—but as it stands, SC offers little control over such expressions. If a representation cannot be manipulated into something that is meaningfully related to $e$ we will struggle to get influence over this representation with SC: Despite being demonstrably irrational, $\zeta(3)$ and the Erdos-Borwein Constant seem to lie beyond the reach of Schanuel's conjecture.
What follows below is a quick and dirty literature review. In this, I can do no better than Ram Murty's chapter on Schanuel's Conjecture in Transcendental Numbers but I am leaving breadcrumbs. Each form in the big list above is a direct result of the references below.
Schanuel's Conjecture Results}
Assuming Schanuel’s Conjecture (SC), the following are algebraically independent over $\mathbb{Q}$:
$$
\left\{
e,\ e^\pi,\ e^e,\ e^i,\ \pi,\ \pi^e,\ \pi^i,\ 2^\pi,\ 2^e,\ \log \pi,\ \log 2,\ \log 3,\ \log \log 2,\ (\log 2)^{\log 3},\ 2^{\sqrt{2}}
\right\}
$$
(Source: Marques & Sondow, 2010)This implies the transcendence of $e + \pi$, $e\pi$, and $q(e,\pi)$ for any rational function $q$. On slide 31 of Waldschimdt, M. we can read that the following are algebraically independent. $\{ e+\pi,e\pi,\pi^{e},e^{\pi^{2}},e^{e},e^{e^{2}},...,e^{e^{e}} ,..., \pi^{\pi},\pi^{\pi^{\pi}},...,\ \log\left(\pi\right),
\log\left(\log\left(2\right)\right),\ \pi\log\left(2\right),\log\left(2\right)\log\left(3\right),2^{\log\left(2\right)},\log\left(2\right)^{\log\left(3\right)},... \}$
I must confess, I don't really have a guess at what the last set of ellipses captures. The previous ones I suspect are $e^{e^n}, {}^n e, {}^n\pi$ respectively.
This shows $\pi 2^\sqrt{2}$ transcendental which was asked about here on MSE.
Exponential Towers + Field-Theoretic Framework
$$
\left\{ e,\ e^e,\ e^{e^e},\ e^{e^{e^e}},\ \dots \right\}
$$
is algebraically independent by repeated application of SC.
Define the tower of fields:
$
E_0 = \overline{\mathbb{Q}}, \quad
E_n = \text{alg. closure of } E_{n-1}(\exp(x) \mid x \in E_{n-1}), \quad
E = \bigcup_{n \geq 0} E_n $
Then:
$
\{ \pi,\ \log \pi,\ \log \log \pi,\ \dots \}
$
is algebraically independent over $E$. (Source: Arizona Winter School, 2008 Note: $\log^4(\pi) \notin \mathbb{R}$—this set contains no reals. More accurate is $\log^4(\pi)\cap\mathbb{R} =\emptyset$.
Consequences for Classical Theorems
SC generalizes Hermite–Lindemann For example: $
\alpha \neq 0 \text{ algebraic} \Rightarrow e^{e^{\dots^{e^\alpha}}} \text{ is SC-transcendental}
$
It also generalizes Gelfond–Schneider:
If $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$ and $\beta \in \overline{\mathbb{Q}} \setminus \mathbb{Q}$, then $\log (\alpha)$ and $\alpha^{\beta}$ are algebraically independent. (Source: Some Consequences of Schanuel’s Conjecture)
Gamma Function Results
SC implies the transcendence of
$
\log(\Gamma(x)) + \log(\Gamma(1-x))
$
for all rational $x$ in $(0,1)$.
(Source: \textit{Transcendental Numbers} by M. Ram Murty and P. Rath)
Personal Explorations
A concrete example: $
2^t + 3^t = 1 \Rightarrow t \text{ is transcendental (by SC)} $ More generally, the solutions to:
$$
\sum_{n=1}^k p_n^t = 1 \quad (\text{where } p_n \text{ are distinct primes})
$$
are SC-transcendental.
Note that $e^e$ and $e^{ei}$ are SC-transcendental, so $\cos(e) = (e^{ei} + e^{-ei}) / 2$ is also transcendental. Trigonometric functions at $e$ are rational functions evaluated at SC-transcendental numbers.
What Schanuel's Conjecture Doesn’t (Seem to) Prove
SC does \textit{not} appear to prove:
$$
(e + 1)^{\sqrt{2}},\quad \sin(\sin(e)),\quad e^{\pi^{e^{\pi^{\pi^e}}}},\quad
$$
For these, something stronger may be needed, such as \textbf{Schanuel Subset Conjecture (SSC)} explored in Marques & Sondow, 2012. SSC \textit{does} imply the transcendence of: $(e + 1)^{\sqrt{2}}$. And more generally, for non-constant $P, Q \in \overline{\mathbb{Q}}[x]$ and any $a \in \overline{\mathbb{Q}} \setminus \{0\}$, SSC implies:$
P(e)^a,
P(\pi)^a,
P(\log 2)^a,
P(\pi)^{Q(\pi)},
P(e)^{Q(e)} $
SC doesn't seem to resolve Gelfond's second conjecture or Gelfond's power tower conjecture. The following forms do not seem to be demonstrably SC-transcendental (This seems unlikely but not known).
$$ e^{\alpha _1 e^{\alpha_2 ^{\dots^{e^{\alpha_n}} }}},\alpha_1^{\alpha_2^{\dots^{\alpha_{n}}}}, $$
Note here that the form above is currently unresolved but the simpler ${}^{n} \alpha$ is SC-transcendental. These unproven extensions are from Gelfond.
References
Marques, D., & Sondow, J. (2010). Some Consequences of Schanuel’s Conjecture. arXiv:1010.6216
AWS 2008 Workshop Group. Some Consequences of Schanuel’s Conjecture. Arizona Winter School
Murty, M.R., & Rath, P. (2013). Transcendental Numbers. Springer. Book — Also available here
Marques, D., & Sondow, J. (2012). The Schanuel Subset Conjecture Implies Gelfond’s Power Tower Conjecture. arXiv:1212.6931
Macintyre, A. (1991). Schanuel’s Conjecture and Free Exponential Rings. Ann. Pure Appl. Logic, 51, 241–246.
Macintyre, A. (2016). Schanuel’s Conjecture and Decidability Problems. Annals of Pure and Applied Logic, 167(10), 830–841. I have not used this inline [yet], but it is an interesting paper. Schanuel's conjecture is also interesting from a theoretical computer science perspective and we can construct forms that are SC-Transcendental. ScienceDirect
Waldschmidt, M. Schanuel’s Conjecture: algebraic independence of transcendental numbers PDF
