We study some anisotropic equations involving the Finsler $N$-Laplace operator $ \\Delta_{H, N},$ with Trudinger-Moser type critical exponential nonlinearity which involves the anisotropic norm. More precisely, we establish the existence of nontrivial solutions to the problem
\\begin{equation}
-\\Delta_{H, N} u = f(x, u),\\;u\\geq0,
\\end{equation}
on a smooth bounded domain $ \\Omega \\subset \\mathbb{R}^N, N \\ge 2$, and to the anisotropic quasilinear Schr\\{o}dinger type problem
\\begin{equation}
-\\Delta_{H, N} u + V(x) |u|^{N-2} u= g(x, u),\\; u\\geq0,
\\end{equation}
in the whole Euclidean space $\\mathbb{R}^N.$ Here $\\\\Delta_{H, N} u = \\text{div} (H(\\nabla u)^{N-1} \\nabla_{\\xi} H(\\nabla u))$ is known as Finsler $N$--Laplacian or anisotropic $N$--Laplacian, $ H: { \\mathbb{R}^N} \\to [0, \\infty)$ is a positively homogeneous Minkowski norm, and $\\nabla_{\\xi}$ denotes the gradient operator with respect to the $\\xi$ variable. The nonlinearities $f$ and $g$ exhibit critical exponential type growth, and the potential $V$ is a continuous function that retains the compactness of the associated Sobolev embedding into a larger class of Lebesgue spaces under suitable assumptions.