About transcendential functions

I have been studing abstract algebra for a while now and I'm familiar with the concept of elements being algebraic or transcendential over certain field (or a ring?). Well anyway, yesterday I faced this interesting post regarding the solving of the equation:
\[
x^x = 25
\]
I have never really given a thought about $x^x$ before, but I discovered that it is a so called transcendential function, meaning "it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction." (wikipedia).

Why this is so interesting? It is because this equation mentioned earlier can't be solved by using algebra, only with numerical methods. Here's couple of other interesting examples of transcendential functions:
$x^{\pi}$
$x^{1/x}$
$c^x$, where $c$ not in ${0, 1}$