Suppose, $u$ solves the equation $$u^u=\pi$$ and $v$ solves the equation $$v\cdot e^v=\pi$$

So, we have $u=e^{W(\ln(\pi))}$ and $v=W(\pi)$. $u$ and $v$ should be the real solutions (in this case, they are unique). If someone can prove that $u$ and $v$ are transcendental, the next step would be to show that every solution is transcendental, but for the moment, the real case is sufficient.

- Are $u$ and $v$ transcendental ?

The Lindemann-Weierstrass-theorem does not help here because $\pi$ is transcendental and $\ln(\pi)$ and $W(\ln(\pi))$ are probably (does anyone know a proof ?) transcendental.

The algdep-command of PARI/GP does neither indicate the algebraicy of $u$ nor the algebraicy of $v$. So, if $u$ or $v$ is algebraic, the minimal polynomial must have high degree or coefficients with a large absolute value.