I know this famous example due to Kollár:
Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover of $E\times E$ branched at a smooth divisor $B\in|3(F_1+F_2)|$, and consider the divisor $A_m=mF_1+(m^2-m+1)F_2-(m-1)\Delta$ for $m>0$. By Nakai-Moishezon it is ample, and thus also $f^*A_m$. But then one can show that $mf^*A_m$ isn't very ample on $X$. Therefore, there doesn't exist a constant $\nu$ on $X$ such that $n A$ is very ample for every ample $A$ on $X$ and every $n\geq \nu$.
I was wondering whether there is an example where we have a smooth projective variety $X$ and a family of ample divisors $\{A_m\}_{m\geq 1}$ such that $mA_m$ has a basepoint for every $m$? I'm sure there must be one, but I couldn't find it.