We consider an extended variant of the classical coupon collector's problem with infinite number of collections. An arriving coupon is placed in the $r^{th}$ collection, $r\ge0$, if $r$ is the smallest index such that the corresponding collection still does not have a coupon of this type. We derive distributional limit theorems for the number of empty spots in different collections at the time when the $0^{th}$ collection was completed, as well as after some delay. We also obtain limiting distributions for completion times of different collections. All main results are given in an ultimate infinite-dimensional form in the sense of distributional convergence in $\mathbb R^\infty$. The main tool in the proofs is convergence of specially constructed point processes.