What are some increasing and always positive functions that is not an exponential function and very simple?
I think it might be very difficult to find a simple function satisfying these two criteria. Here is my informal reasoning. The function $f$ is increasing and has a lower bound at zero. Suppose $f(x)\to 0$ when $x\to-\infty$ This implies that $f'(x)$ cannot be large when $x\to -\infty$. It should be the case that $f'(x)\to0^+$ when $x\to-\infty$, which suggests that when $x$ is moving from $-\infty$ to $0$, the big picture of $f'(x)$ must be increasing. Repeating this logic we have $f^{(n)}(x)$ to be increasing and $f^{(n)}(x)\to 0$ when $x\to -\infty$. This seems like an exponential function.
One cadidate is the arctan function, but trigonometry functions are just a form of exponential function.
Positive linear combinations of exponential functions (and constants) are also in this category, as are products and roots of such functions, so there can't be too narrow a description. There are also hyperbolas such as $x + \sqrt{x^2+1}$, and thus we get further combinations like $\sqrt[7]{x + \sqrt{x^2+1} + 2e^{3x} + 4e^{5x} + 6}$ and $(x + \sqrt{x^2+1})(e^x+2)(\arctan(3x)+4)$. Probably there are many others as well.