Minimal surfaces with rapid area growth

We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into ${\mathbf{R}}^{\mathbf{4}}$ that yields a stable minimal surface, while the second is a proper immersion into ${\mathbf{R}}^{\mathbf{3}}$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.