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 $ζ_\infty(t)=1-e^{-t}$ of the complete collection, so we call $ζ_\infty$ the (asymptotic) \emph{completion curve}. In this paper, for $ν>0$, we address the asymptotic shape $ζ(ν,.) $ of the completion curve under the condition $T_n\leq \left( 1+ν\right) n$, 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 Koršunov.