Most approaches to multi-talker overlapped speech separation and recognition assume that the number of simultaneously active speakers is given, but in realistic situations, it is typically unknown. To cope with this, we extend an iterative speech extraction system with mechanisms to count the number of sources and combine it with a single-talker speech recognizer to form the first end-to-end multi-talker automatic speech recognition system for an unknown number of active speakers. Our experiments show very promising performance in counting accuracy, source separation and speech recognition on simulated clean mixtures from WSJ0-2mix and WSJ0-3mix. Among others, we set a new state-of-the-art word error rate on the WSJ0-2mix database. Furthermore, our system generalizes well to a larger number of speakers than it ever saw during training, as shown in experiments with the WSJ0-4mix database.