Let $X(t)$ be a stationary Gaussian process with continuous sample paths. The behavior of $|X(t)|$ as $t \rightarrow \infty$ is considered. In particular, conditions on the spectrum of the process are given which determine whether $\lim \sup_{t\rightarrow\infty}|X(t)|/(\log t)^{\frac{1}{2}} = \operatorname{Const.} > 0$. These conditions are complete except when the spectrum of the process is continuous-singular. The main concern of this paper is to study the asymptotic behavior of some specific examples of $X(t)$ with continuous-singular spectra. Many examples are given showing the asymptotic behavior of stationary Gaussian processes with discrete spectra and their indefinite integrals.