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Recurrent Neural Cascades (RNCs) are the recurrent neural networks with no cyclic dependencies among recurrent neurons. This class of recurrent networks has received a lot of attention in practice. Besides training methods for a fixed architecture such as backpropagation, the cascade architecture naturally allows for constructive learning methods, where recurrent nodes are added incrementally one at a time, often yielding smaller networks. Furthermore, acyclicity amounts to a structural prior that even for the same number of neurons yields a more favourable sample complexity compared to a fully-connected architecture. A central question is whether the advantages of the cascade architecture come at the cost of a reduced expressivity. We provide new insights into this question. We show that the regular languages captured by RNCs with sign and tanh activation with positive recurrent weights are the star-free regular languages. In order to establish our results we developed a novel framework where capabilities of RNCs are assessed by analysing which semigroups and groups a single neuron is able to implement. A notable implication of our framework is that RNCs can achieve the expressivity of all regular languages by introducing neurons that can implement groups.