What do you mean by analytic? Analytic on the real line, or entire. The method you suggest can be used to obtain an entire example. Let $F$ be a Gaussian. Consider a series $$\sum a_nF(z-c_n)$$ with positive $a_n$ and $c_n$. If $a_n\to+\infty$ does not grow fast, but $c_n$ does grow fast enough, the series converges on every compact in the complex plane, and gives you an entire function with the properties you stated.

What do you mean by analytic? Analytic on the real line, or entire. The method you suggest can be used to obtain an entire example. Let $F$ be a Gaussian. Consider a series $$\sum a_nF(b_n(z-c_n))$$ with positive $a_n, b_n$ and $c_n$. If $a_n\to+\infty$ does not grow fast, but $c_n$ does grow fast enough, the series converges on every compact in the complex plane, and gives you an entire function with the properties you stated.

EDIT. I corrected according to R. Israel suggestion. With $b_n=1$ and $a_n\to\infty$ it will not be in $L^1$.

In fact, Carleman's theorem says that for every continuous function $\phi$ on the real line, and every positive continuous function $\epsilon$ on the real line, one can find an entire function $f$ such that $|f(x)-\phi(x)|<\epsilon(x)$.