Let $\Omega \subset {\mathbb{R}}^{n}$, $n\ge 2$, be a bounded connected domain of the class ${C}^{1,\theta}$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in {W}^{1}{L}^{\Phi}\left(\Omega \right),\phantom{\rule{1.0em}{0ex}}-div\left({\Phi}^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi}^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\phantom{\rule{1.0em}{0ex}}\text{in}\Omega ,\frac{\partial u}{\partial \mathbf{n}}=0\phantom{\rule{1.0em}{0ex}}\text{on}\partial \Omega ,$$
where $\Phi $ is a Young function such that the space ${W}^{1}{L}^{\Phi}\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...