setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic microwave architecture allows to overcome this difficulty. As an application, we show that this architecture allows to generate a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states, and opens the quest for continuous-variable algorithms based on a few repetitions of elementary gates from the continuous-variable universal set.