Let $X(t), t\in \mathcal{T}$ be a centered Gaussian random field with variance function $σ^2(\cdot)$ that attains its maximum at the unique point $t_0\in \mathcal{T}$, and let $M(\mathcal{T}):=\sup_{t\in \mathcal{T}} X(t)$. For $\mathcal{T}$ a compact subset of $\R$, the current literature explains the asymptotic tail behaviour of $M(\mathcal{T})$ under some regularity conditions including that $1- σ(t)$ has a polynomial decrease to 0 as $t \to t_0$. In this contribution we consider more general case that $1- σ(t)$ is regularly varying at $t_0$. We extend our analysis to random fields defined on some compact $\mathcal{T}\subset \R^2$, deriving the exact tail asymptotics of $M(\mathcal{T})$ for the class of Gaussian random fields with variance and correlation functions being regularly varying at $t_0$. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.