Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0-x|_g^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0-x|_g$ denotes distance induced by $g$.
In particular there is no examples which you are looking for.
See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.