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U-Net is a widely adopted neural network in the domain of medical image segmentation. Despite its quick embracement by the medical imaging community, its performance suffers on complicated datasets. The problem can be ascribed to its simple feature extracting blocks: encoder/decoder, and the semantic gap between encoder and decoder. Variants of U-Net (such as R2U-Net) have been proposed to address the problem of simple feature extracting blocks by making the network deeper, but it does not deal with the semantic gap problem. On the other hand, another variant UNET++ deals with the semantic gap problem by introducing dense skip connections but has simple feature extraction blocks. To overcome these issues, we propose a new U-Net based medical image segmentation architecture R2U++. In the proposed architecture, the adapted changes from vanilla U-Net are: (1) the plain convolutional backbone is replaced by a deeper recurrent residual convolution block. The increased field of view with these blocks aids in extracting crucial features for segmentation which is proven by improvement in the overall performance of the network. (2) The semantic gap between encoder and decoder is reduced by dense skip pathways. These pathways accumulate features coming from multiple scales and apply concatenation accordingly. The modified architecture has embedded multi-depth models, and an ensemble of outputs taken from varying depths improves the performance on foreground objects appearing at various scales in the images. The performance of R2U++ is evaluated on four distinct medical imaging modalities: electron microscopy (EM), X-rays, fundus, and computed tomography (CT). The average gain achieved in IoU score is 1.5+-0.37% and in dice score is 0.9+-0.33% over UNET++, whereas, 4.21+-2.72 in IoU and 3.47+-1.89 in dice score over R2U-Net across different medical imaging segmentation datasets.