A novel channel estimation framework is presented for orthogonal frequency division multiplexing (OFDM) system that operates with low-precision analog-to-digital converters (ADCs). The framework is based on concatenated minimum mean square error (MMSE) estimation, which consists of an inner and an outer MMSE estimation blocks. The outer MMSE estimation finds the compound signal, i.e., the unknown channel multiplied by a pilot signal, from the nonlinear distortion of the received signal by the quantization and its mean square error (MSE). Using the estimated compound signal and its MSE, the inner MMSE estimation estimates the desired signal, i.e., the unknown channel value, assuming the resulting estimation error of the outer MMSE block follows the Gaussian distribution. One major finding is that the proposed framework is analytically tractable for quantifying the effective quantization error, unlike the widely-used Bussgang-based approach. From simulations, it is shown that the proposed channel estimation framework provides a significant gain over conventional Bussgang-based methods using the approximated covariance matrix of the quantization error.