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In frequency division duplexing systems, the base station (BS) acquires downlink channel state information (CSI) via channel feedback, which has not been adequately investigated in the presence of RIS. In this study, we examine the limited channel feedback scheme by proposing a novel cascaded codebook and an adaptive bit partitioning strategy. The RIS segments the channel between the BS and mobile station into two sub-channels, each with line-of-sight (LoS) and non-LoS (NLoS) paths. To quantize the path gains, the cascaded codebook is proposed to be synthesized by two sub-codebooks whose codeword is cascaded by LoS and NLoS components. This enables the proposed cascaded codebook to cater the different distributions of LoS and NLoS path gains by flexibly using different feedback bits to design the codeword structure. On the basis of the proposed cascaded codebook, we derive an upper bound on ergodic rate loss with maximum ratio transmission and show that the rate loss can be cut down by optimizing the feedback bit allocation during codebook generation. To minimize the upper bound, we propose a bit partitioning strategy that is adaptive to diverse environment and system parameters. Extensive simulations are presented to show the superiority and robustness of the cascaded codebook and the efficiency of the adaptive bit partitioning scheme.