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Let $G$ be a connected graph. The multiplicative Zagreb eccentricity indices of $G$ are defined respectively as ${bf Pi}_1^*(G)=prod_{vin V(G)}varepsilon_G^2(v)$ and ${bf Pi}_2^*(G)=prod_{uvin E(G)}varepsilon_G(u)varepsilon_G(v)$, where $varepsilon_G(v)$ is the eccentricity of vertex $v$ in graph $G$ and $varepsilon_G^2(v)=(varepsilon_G(v))^2$. In this paper, we present some bounds of the multiplicative Zagreb eccentricity indices of Cartesian product graphs by means of some invariants of the factors and supply some exact expressions of ${bf Pi}_1^*$ and ${bf Pi}_2^*$ of some composite graphs, such as the join, disjunction, symmetric difference and composition of graphs, respectively.