Gelfond's constant is the transcendental number $e^\pi$ which is approximately equal to:
$$ 23.140692632... $$
Are there any known similar constants (i.e. referred to in literature or used in specific contexts) of the form $e^c$ where:
- $c=\pi^2, \pi^3, \ldots$?
- $c=\pi k$, where $k$ is a constant (such as $\sqrt{163}$ for Ramanujan's constant)?
- $c=\pi f(x)$ for some $x$?
Reading through Brownawell and Waldschmidt, they propose results in these directions which do not rely on the so-called Schanuel's Conjecture. The references are as follows
Solution du Huitième Problème de Schneider,
The algebraic independence of certain numbers related by the exponential function
These papers independently prove results along the following lines.
Let $\alpha$, $\beta$, and $\gamma$ be nonzero complex numbers with $\alpha$ and $\beta$ both irrational. If $e^\gamma$ and $e^{\alpha\gamma}$ are both algebraic numbers, then at least two of the numbers $$\alpha, \beta, \gamma, e^{\beta\gamma}, e^{\alpha\beta\gamma}$$ are algebraically independent over $\mathbb{Q}$.
This raises several interesting consequences:
Taking $\alpha=\beta=e^{-1}, \gamma=e^2$, we see that at least one of $e^e$ and $e^{e^2}$ must be transcendental. This was conjectured by Schneider.
Taking $\alpha=\beta=\gamma$, we see that given any nonzero complex number $\alpha$, at least one of the numbers $e^{\alpha}, e^{\alpha^2}, e^{\alpha^3}$ must be transcendental.
Taking $\alpha = \beta = i/\pi, \gamma=\pi^2$, we see that at least one of the following holds: (i) $e^{\pi^2}$ is transcendental, or (ii) $e$ and $\pi$ are algebraically independent.
So as a partial answer to this question, at least one of $e\pi$ and $e^{\pi^2}$ is transcendental.
Whilst this does not answer fully your question, it may be of decent enough reference to continue your quest for showing the transcendental nature of some of the reuslts you seek.