# Other Algebraically Independent Transcendentals

I was thinking about a incomplete answer I gave earlier today to a interesting question made by the user lurker. The question was about wheter or not the sum or the product of two transcendental numbers in $\mathbb{C}/\mathbb{Q}$ could result on a algebraic number in the same set. A infinite amount of trivial examples can be shown by manipulating the same transcendental, like shown in the answers and in the comments, but I was intrigued by what I believed being the question's true aim, the non-trivial results.
After fiddling around with some algebra to try and understand the nature of transcendental extensions, then proceed to read about it during the day, I reached the known result that it depends on wheter or not the two given transcendental numbers are algebraically independent. If they are, then we cannot generate a 'non trivial algebraic' transformation between them that yields a algebraic number (i.e. using only the four basic operations). It's just a question of stating a polynomial with rational coefficients whose roots are the given transcendental numbers.
Now to the real question. From Nesterenko's results [1996] we know the independence of $\pi$, some values of the Gamma $\Gamma$ function and some powers of $e^\pi$. From the unproved Schanuel's conjecture follows that $\pi$ and $e$ are independent.

1. Are there examples of more notable (conjectured or known) algebraically independent transcendental numbers?
2. Are there a pair of transcendental numbers, maybe notable constants, whose algebraical dependence has been proved or conjectured?