There are some results that i found
$1.$ if $a$ is an algebraic number other than $0$ and $1$ and $b$ is irrational algebraic then $a^{b}$ is transcendental like $2^{\sqrt{5}},3^{\sqrt{7}}$etc.
$2.$ $e^{a}$ is transcendental for algebraic number $a.$
Now i am searching other results like $(b)^{b}$ where $b$ is transcendental. Is it algebraic or transcendental? like $\pi^{\pi},e^{e}$etc. Thanks in advance.
The set of algebraic numbers is countable but the set of transcendental reals is uncountable. And any real interval of positive length is an uncountable set . It follows that $b^b$ is transcendental for "almost all" transcendental $b$, that is, for all but countably many $b$. I don't know whether the status of $b^b$ is known for any specific cases. And if there is a good abbreviation for "transcendental" my fingers will be grateful.The results you quoted :1. is called the Gel'fond Theorem.And 2.( for $a\ne 0$) is a special case of the Hermite-Lindeman Theorem.