Why is the exponential function not in the subspace of all polynomials?

If $p$ is a polynomial of degree $n$, then the $n$th derivative of $p$ is constant. Note that the $n$th derivative of $e^x$ is $e^x$. Now all you have to do is prove that $e^x$ is not constant.

The function $e^x$ is not in $\text{span}\{1, x,x^2, \dots \}$ because it is no finite linear combination of basis elements (but a countable one). What is true is that $$e^x \in \overline{\text{span}\{1, x,x^2, \dots \}}$$ is in the closure because you can find a sequence in $\text{span}\{1, x,x^2, \dots \}$ which converges to $e^x$. I hope it helps you :)

$\mathrm{span}(A)$ is the set of finite linear combinations of terms from $A$. Infinite sums require notions of limits and bring up issues of convergence radii (there are plenty of infinite polynomial that converge only at a single point).