Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensated by coherent input drives. Here we investigate a system consisting of three transmission line resonators, where the two outer ones are driven by coherent input sources and the central resonator interacts with a superconducting qubit. Whereas a low excitation number regime of such a device has been considered previously with a numerical integration, we here specifically address the high excitation density regime. This is of particular interest as intra cavity fields might undergo a transition from low excitation number quantum fields to high amplitude classical fields when increasing the input drives. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov expansion in quantum fluctuations which can be understood as an approximation for weak nonlinearities. As the second method we introduce a combination of mean-field decoupling for the photon tunneling with an exact approach to a driven Kerr nonlinearity which can be understood as an approximation for low tunneling rates.