To achieve the more promising passive beamforming gains in the double-intelligent reflecting surface (IRS) assisted system over the conventional single-IRS system, channel estimation is practically indispensable but also a more challenging problem to tackle, due to the presence of not only the single- but also double-reflection links that are intricately coupled. In this paper, we propose a new and efficient channel estimation scheme for the double-IRS assisted uplink multiple-input multiple-output (MIMO) communication system to resolve the cascaded channel state information (CSI) of both its single- and double-reflection links. First, for the single-user case, the higher-dimensional double-reflection channel is efficiently estimated at the multi-antenna base station (BS) with low training overhead by exploiting the fact that its cascaded channel coefficients are scaled versions of those of a lower-dimensional single-reflection channel. Then, the proposed channel estimation scheme is extended to the multi-user case, where given an arbitrary user's cascaded channel estimated as in the single-user case, the other users' cascaded channels are scaled versions of it and thus can be estimated with reduced training overhead. Simulation results verify the effectiveness of the proposed channel estimation scheme as compared to the benchmark scheme.