I was recently exploring the set of equations such that $\frac{d^n}{dx^n}f(x)=f^{-1}(x)$.
Through some not too difficult algebra, I found the solution $f(x)=ax^b$ where $a=\frac{(b-n)!}{b!}^{\frac{1}{b-n+1}}$ and $b=\frac{1}{2}(n+\sqrt{n^2+4})$.
I then noticed that the intersection between $y=x$ and $y=\lim_{n \rightarrow 0}\left(\frac{d^n}{dx^n}f(x)\right)$ happens at the point $(e^{1-\gamma},e^{1-\gamma})$. I have verified this identity to at least 6 digits using desmos, but I have no idea how to prove this identity. Any help? Attached is the desmos sheet with a bunch of the work. https://www.desmos.com/calculator/r2qfcqlpi9