Traditional non-interferometric quantitative phase imaging (QPI) methods often face challenges in realizing rapid and accurate imaging of large-phase samples, mainly due to slow convergence and dependence on object approximation models. In this Letter, we propose a new, to the best of our knowledge, non-interferometric QPI approach that leverages iterative Kramers-Kronig (KK) relations, named iKK-QPI, to achieve high-accuracy quantitative measurement of objects with large-phase values. In the current KK relations reconstruction framework, we impose real-part constraints on the cepstrum, breaking the restriction of weak scattering condition. With only a few iterations, iKK-QPI extends the phase range that can be reliably retrieved by non-interferometric QPI, exceeding the first-order Born and Rytov approximations. The capability of iKK-QPI is demonstrated by imaging a microlens array and COS-7 cells. We accurately reconstruct objects with large-phase ranges 6 rad (error <
± 5%), three times that of the KK relations-based method, opening up the possibility for non-interferometric QPI to measure biological and industrial samples with large-phase features.