Recurrent Neural Networks (RNNs) are a popular computational tool in neuroscience. Their architecture is inspired by biology, namely the abundance of lateral connections in cortex. Because they are universal approximators, they can be trained to match behavior in common neuroscience tasks, and their hidden state can be compared to neural recordings. Despite the biological inspiration, many features of real neurons are abstracted away (e.g. morphology, neuromodulators, excitability dynamics, short term plasticity). While many of these properties were included as add-ons to standard RNN models, the basic architecture of linear summation of inputs has remained almost consent. Here, we challenge this architectural assumption and ask what are the implications of a three-body, quadratic, interaction in RNNs. Three-body interaction can arise from several biophysical properties: dendritic nonlinearities, where inputs on the same branch gate each other; incorporating glia cells into the network; the effect of neuromodulators on synaptic transmission. Furthermore, outside neuroscience, gene expression networks are often modeled with RNN-like dynamics. Dimerization of transcription factors, however, results in a pure three-body interaction in this scenario. We thus propose the Three-Body Recurrent Neural Network (TBRNN) model as a novel object of study. First, we show that this class of models serves as a universal approximator for any open dynamical system, a nontrivial theorem attributable to the inherent pure quadratic interactions within the system. Next, we show how the concept of low rank RNN can be extended to this case, both to infer connectivity and to design functionality. Finally, we show how qualitatively different solutions emerge when TBRNNs solve the same tasks as RNNs. Altogether, our work expands the space of dynamical models in neuroscience while also allowing other fields of biology to benefit from insights developed in neuroscience.