For a while I've been interested in functions with fixed points (if I'm correct, functions for which $f(x)=x$), and while playing around with exponential functions of the form $y =a^x$, I noticed that while $y=1.5^x$ doesn't intercept $y=x$ anywhere, $y=1.4^x$ did (at $x=1.887$ and $x=4.41$). So I started fiddling around with the base, and so far it has led me to $y=1.4445043755045700040184897261497^x$, which intercepts $y=x$ approximately at $x=2.652$ and $x=2.787$.
What I want to know is, is this approximately the largest base with nonzero fixed points? If so, is there any significance to that number, or these fixed points?
You are solving $$x=a^x$$ equivalently $$x^{1/x}=a$$ or $$\frac{\ln x}x=\ln a.$$ So we want the maximum of $f(x)=(\ln x)/x$ over $x>1$. Now $$f'(x)=\frac{1-\ln x}{x^2}$$ so the maximum is at $x=e$ giving $$a=e^{1/e}.$$