  In the coupon collector problem with $n$ items, the collector needs a random
number of tries $T_n\simeq n\ln n$ to complete the collection. Also, after $nt$
tries, the collector has secured approximately a fraction
$\zeta_\infty(t)=1-e^{-t}$ of the complete collection, so we call
$\zeta_\infty$ the (asymptotic) \emph{completion curve}. In this paper, for
$\nu>0$, we address the asymptotic shape $\zeta (\nu,.) $ of the completion
curve under the condition $T_n\leq \left( 1+\nu \right) n$, i.e. assuming that
the collection is \emph{completed unlikely fast}. As an application to the
asymptotic study of complete accessible automata, we provide a new derivation
of a formula due to Kor\v{s}unov.
