Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.
In particular there is no examples which you are looking for.
See Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology.