In this paper, we investigate the channel estimation challenge in reconfigurable intelligent surface (RIS)-aided near-field communication systems. Current channel estimation techniques require substantial pilot overhead and computational complexity, especially when the number of RIS elements is extremely large. To address this issue, we introduce a two-timescale channel estimation strategy that leverages the asymmetric coherence times of both the RIS-base station (BS) channel and the User-RIS channel. We derive a time-scaling property indicating that, for any two effective channels within the longer coherence time, one effective channel can be represented as the product of a vector, termed the small-timescale effective channel, and the other effective channel. By integrating the estimated effective channel from the initial time block with observations from our piecewise beam training, we present an efficient method for estimating subsequent small-timescale effective channels. We theoretically verify the efficacy of the proposed RIS design and demonstrate, through simulations, that our channel estimation method outperforms existing methods in pilot overhead and computational complexity across various realistic channel models.