The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty$. In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that $limsup\left|\right|Tⁿ\left|\right|/n^{1-\gamma ₀}=0.$ In this note we construct an example of a nonpositive L¹ operator with the highest possible...