Estimating time-frequency domain masks for speech enhancement using deep learning approaches has recently become a popular field of research. In this paper, we propose a mask-based speech enhancement framework by using concatenated identical deep neural networks (CI-DNNs). The idea is that a single DNN is trained under multiple input and output signal-to-noise power ratio (SNR) conditions, using targets that provide a moderate SNR gain with respect to the input and therefore achieve a balance between speech component quality and noise suppression. We concatenate this single DNN several times without any retraining to provide enough noise attenuation. Simulation results show that our proposed CI-DNN outperforms enhancement methods using classical spectral weighting rules w.r.t. total speech quality and speech intelligibility. Moreover, our approach shows similar or even a little bit better performance with much fewer trainable parameters compared with a noisy-target single DNN approach of the same size. A comparison to the conventional clean-target single DNN approach shows that our proposed CI-DNN is better in speech component quality and much better in residual noise component quality. Most importantly, our new CI-DNN generalized best to an unseen noise type, if compared to the other tested deep learning approaches.