I found out that: $$\frac{\int \Pi(x)H(x)dx-\Pi(x)}{\int \Pi(x)dx}=\gamma$$ Where $H(x)$ is the $x$th harmonic number, $\Pi(x)$ is the analytic continuation of $z!$, and $\gamma$ is the Euler-Mascheroni constant. Every function here is not elementary, so I thought that a number written in terms of functions that aren't elementary is transcendental. But of course this is just an assumption. Are there any results similar to my guess?
Edit: To prove this identity, start with the fact that: $$\Pi'(x)=\Pi(x)\left(-\gamma+H(x)\right)$$ Take the integral of both sides: $$\Pi(x)=\int\Pi(x)(-\gamma+H(x))dx=-\gamma\int\Pi(x)dx+\int H(x)\Pi(x)dx$$ Solve for $\gamma$ to get the orginal identity. Also, I replaced $\Gamma(x+1)$ with the equivalent $\Pi(x)$