Can a transcendental number like $e$ or $\pi$ be compressed as not algorithmically random?
The problem is in your poor definition of &quot;algorithmically random number&quot; as applied to irrational numbers. In particular:

has the same length (number of bits) as the number itself.

has no meaning if the number is of unbounded length.
Your Wikipedia link gives better definitions, which don't have this problem. For example (and paraphrasing formatting):

Kolmogorov complexity [...] can be thought of as a lower bound on the algorithmic compressibility of a finite sequence (of characters or binary digits). It assigns to each such sequence $w$ a natural number $K(w)$ that, intuitively, measures the minimum length of a computer program (written in some fixed programming language) that takes no input and will output $w$ when run. Given a natural number $c$ and a sequence $w$, we say that $w$ is $c$-incompressible if $K(w) \geq |w| - c$.
An infinite sequence $S$ is Martin-Löf random if and only if there is a constant $c$ such that all of $S$'s finite prefixes are $c$-incompressible.

This is a test passed by $\sqrt{e}$ by setting $c$ a bit larger than the program to generate $\sqrt{e}$ and including in it the length to generate.