Let $f\in C^\infty(\mathbb{R})$ be a smooth and positive function with support in $[-1,1]$, satisfying $\int f(t)\,dt = 1$. Define $g_1 = f$ and $g_{k+1} = f\ast g_k$. That is, $g_k$ is the $k$-fold convolution of $f$ with itself.
Is there necessarily some $N$ large enough (depending on $f$) such that $g_k$ is log-concave for all $k>N$?
By the central limit theorem, $g_k$ will of course eventually be well approximated by a Gaussian distribution, which is log-concave. But log-concavity is not stable under perturbations, so this alone does not make for an argument.
EDIT: Some further evidence for this hypothesis is the fact that the heat flow of a compactly supported function is eventually log-concave. Moreover, I have now tested this numerically and it appears to hold.