Sorry if I wasn't clear. f(x) in my example is not e^x
f(x) = 1/(1+e^(-x))
If you raise this function to higher and higher powers, it will look more and more like a step function and have a super-exponential growth phase on a log-chart (in the limit, it will look like a step function on the log chart as well as the linear chart)
Just plotted it, I stand corrected!
For high enough powers, it will look like a step-function on a linear plot but a "corner" shape (i.e., exactly vertical and then exactly horizontal) on a log plot, my bad!
So yes, it will never look super-exponential. Sorry!
If I could say one thing in my defense though, if you modify f(x) slightly by adding a constant epislon (a teency weency "intrinsic value", say epsilon = 0.001), then it looks like a step function on both the linear and log plots for very high powers.
f(x) = 1/(1+e^(-x))^n + epislon
where n -> infinity
