Dyson-Schwinger equations provide one of the most powerful non-perturbative approaches to quantum field theories. The quartic analogue of the Kontsevich model is a toy model for QFT in which the tower of Dyson-Schwinger equations splits into one non-linear equation for the planar two-point function and an infinite hierarchy of affine equations for all other functions. The non-linear equation admits a purely algebraic solution, identified through insight from perturbation theory. The affine equations turn out to be affiliated with (and solved by) a universal structure in complex algebraic geometry: blobbed topological recursion. As such they connect to the geometry of the moduli space of complex curves.