Is there an elementary function function $f(x)$ with the following properties:
- $f(x)$ is defined and infinitely differentiable on all of $\mathbb{R}$
- $f(x)$ is bounded
- $f^\prime(x)$ is nonnegative and unbounded
If we don't require $f(x)$ to be an elementary function, then we can take $$f(x)=\int_0^x\frac{t^2}{1+t^8\sin^2(\pi t)}\,dt$$ (the integrand is nonnegative, unbounded, but has finite area, desmos link).
This question can also be rephrased in terms of an unbounded probability mass function with elementary antiderivative.