We construct several examples of Hilbertian operator spaces with few completely bounded maps. In particular, we give an example of a separable $1$-Hilbertian operator space $X_0$ such that, whenever $X'$ is an infinite dimensional quotient of $X_0$, $X$ is a subspace of $X'$, and $T : X \raw X'$ is a completely bounded map, then $T = \lambda I_{X} + S$, where $S$ is compact Hilbert-Schmidt and $||S||_2/16 \leq ||S||_{cb} \leq ||S||_2$. Moreover, every infinite dimensional quotient of a subspace of $X_0$ fails the operator approximation property. We also show that every Banach space can be equipped with an operator space structure without the operator approximation property.