Non-Runge Fatou-Bieberbach Domains in Stein Manifolds with the Density Property

Let $X$ be a Stein manifold with the density property. We present methods of constructing two kinds of non-Runge Fatou-Bieberbach domains in $X$, which by definition are proper open subsets of $X$ biholomorphic either to ${\mathbb{C}}^n$ or to $X$. For both kinds, we provide examples where our methods apply.