Photon superbunching, which occurs when the second-order correlation satisfies $g^{(2)}> 2$, is typically associated with strong optical nonlinearities or collective multi-photon emission processes. We predict that extreme superbunching can also arise in systems of weakly-nonlinear photonic cavities, via the creation of a squeezed vacuum through interference engineering by fine-tuning inter-cavity couplings and drive parameters. We present numerical calculations indicating that a system of four photonic resonators containing representative Kerr media can achieve $g^{(2)}(0) = 135$ with a $80\,\text{kHz}$ emission rate. Unlike earlier superbunching schemes, this mechanism is highly compatible with integrated photonic platforms constructed using conventional optical media.