Are there an infinite amount of functions where the limit as z approaches infinity of f_prime^z(x) = 0?

Question

Are there an infinite amount of functions that you can take the derivative of an infinite amount of times without ever being equal to zero?

I'll try to clarify- if you take the derivative of f(x) an infinite amount of times and it doesn't equal zero, is there an infinite amount of functions f(x) could be? (A proof would be cool!)

f(x) = e^x is a possibility, as it is it's own derivative, and I believe the trigonometric functions work? Are there others?

Edit: Of different forms! Obviously the functions e^x, 2e^x, 3e^x... continues for infinity and all fit the description.

Answer 1

It might be more helpful to consider the complement of the question: what functions do not differentiate to zero eventually:

Only real polynomials eventually differentiate to zero.

Helpful related query: When do polynomials eventually differentiate to zero?

Answer 2

Do you consider the set of all polynomials to contain an infinite number of different forms? Because $e^xp(x)$ (where $p(x)$ is a non-zero polynomial) never vanishes no matter how many times you differentiate it. (You can easily prove this as the highest order term of $p$ stays the same.)