Playing with some algebra using only the assumption that the algebraic numbers are closed under algebraic functions, I discovered it's easy to show that for any pair of transcendental numbers $t_1,t_2\in \mathbb{T}$ and the algebraic numbers $\overline{\mathbb{Q}}$, if we define the set $X$ by:
$X=\{t_1\times(t_2+\overline{q}): \overline{q}\in \overline{\mathbb{Q}}\}$
then $\lvert X \cap \overline{\mathbb{Q}}\rvert\leq1$
This implies for example if we take $\{e\times (\pi+q): q\in\overline{\mathbb{Q}}\}$, at most one element of this infinite set is algebraic - giving some measure of just how unlikely it is that $e\times\pi$ is algebraic.
I thought this was kind of interesting and I was able to extend it in various ways such as showing $e+\pi$ is also almost certainly transcendental (since $\lvert\{t_1+\overline{q}t_2: q\in\overline{\mathbb{Q}}\}\cap\overline{\mathbb{Q}}\rvert\leq1$).
I'm not an advanced mathematician; I just like to work things out for myself so I often discover things like this which are already known, which I presume is the case here. So I presume this is a special case of some theorem already in use. What theorem is it, and what field of maths is this? I'm guessing it's something like transcendental extensions of algebraic fields.