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This work is concerned with the following Klein-Gordon-Maxwell system: −Δu+V(x)u−(2ω+ϕ)ϕu=f(u),x∈R2,Δϕ=(ω+ϕ)u2,x∈R2,\left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\end{array}\right. where ω>0\omega \gt 0 is a constant, u,ϕ:R2→Ru,\phi :{{\mathbb{R}}}^{2}\to {\mathbb{R}}, V∈C(R2,R)V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}), and f∈C(R,R)f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) obeys exponential critical growth in the sense of the Trudinger-Moser inequality. We give some new sufficient conditions on ff, specifically related to exponential growth, to obtain the existence of nontrivial solutions. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level.