In this paper, the carrier frequency offset (CFO) estimation bound is presented to assess the CFO estimation performance of an orthogonal frequency division multiplexing (OFDM)-based single relay network over multipath channels, for which an amplify-and-forward (AF) protocol is adopted. We theoretically derive the Cramer-Rao bounds (CRBs) of the CFO estimation in multipath channels. Unlike the analysis in flat fading channels where the CFOs of the two hops can be combined together as a single estimation parameter, the CFOs of the source-relay (S-R) and the relay-destination (R-D) links are required to be evaluated separately in multipath channels. The first-order Taylor series expansion is applied to simplify the calculation of the inverse of the covariance matrix which reflects the correlation of multipath channels and relay noise. In addition, we derive the modified CRB (MCRB) of the CFO estimation to get more insight into the influence of the system parameters on the CRB performance. Through computer simulations, we evaluate the impact of multipath channels, the operating signal-to-noise ratio (SNR), and the number of effective multipaths on the CFO estimation error in the dual-hop relay transmission.