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 $|f(x)-\phi(x)|<\epsilon(x)$.