The ring of values of $E$-functions is not a field

Let $\mathbf{E}$ be the ring of values taken at algebraic points by $E$-functions. The exponential function and the Bessel function $J_0$ are classical examples of $E$-functions. We prove that $\mathbf{E}$ is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of $\mathbf{E}$ explicity, including $e+1$, $J_0(1)$, and ${}_1{F}_1(a;b;1)$ for rational parameters $a$ and $b$ such that $0