  For each ordinal $0\leqslant \xi\leqslant \omega_1$, we introduce the notion
of a $\xi$-completely continuous operator and prove that for each ordinal $0<
\xi< \omega_1$, the class $\mathfrak{V}_\xi$ of $\xi$-completely continuous
operators is a closed, injective operator ideal which is not surjective,
symmetric, or idempotent. We prove that for distinct $0\leqslant \xi,
\zeta\leqslant \omega_1$, the classes of $\xi$-completely continuous operators
and $\zeta$-completely continuous operators are distinct. We also introduce an
ordinal rank $\textsf{v}$ for operators such that $\textsf{v}(A)=\omega_1$ if
and only if $A$ is completely continuous, and otherwise $\textsf{v}(A)$ is the
minimum countable ordinal such that $A$ fails to be $\xi$-completely
continuous. We show that there exists an operator $A$ such that
$\textsf{v}(A)=\xi$ if and only if $1\leqslant \xi\leqslant \omega_1$, and
there exists a Banach space $X$ such that $\textsf{v}(I_X)=\xi$ if and only if
there exists an ordinal $\gamma\leqslant \omega_1$ such that
$\xi=\omega^\gamma$. Finally, prove that for every $0<\xi<\omega_1$, the class
$\{A\in \mathcal{L}: \textsf{v}(A) \geqslant \xi\}$ is $\Pi_1^1$-complete in
$\mathcal{L}$, the coding of all operators between separable Banach spaces.
This is in contrast to the class $\mathfrak{V}\cap \mathcal{L}$, which is
$\Pi_2^1$-complete in $\mathcal{L}$.
