In this paper, we propose a movable antenna (MA) enhanced scheme for wireless powered mobile edge computing (WP-MEC) system, where the hybrid access point (HAP) equipped with multiple MAs first emits wireless energy to charge wireless devices (WDs), and then receives the offloaded tasks from the WDs for edge computing. The MAs deployed at the HAP enhance the spatial degrees of freedom (DoFs) by flexibly adjusting the positions of MAs within an available region, thereby improving the efficiency of both downlink wireless energy transfer (WPT) and uplink task offloading. To balance the performance enhancement against the implementation intricacy, we further propose three types of MA positioning configurations, i.e., dynamic MA positioning, semi-dynamic MA positioning, and static MA positioning. In addition, the non-linear power conversion of energy harvesting (EH) circuits at the WDs and the finite computing capability at the edge server are taken into account. Our objective is to maximize the sum computational rate (SCR) by jointly optimizing the time allocation, positions of MAs, energy beamforming matrix, receive combing vectors, and offloading strategies of WDs. To solve the non-convex problems, efficient alternating optimization (AO) frameworks are proposed. Moreover, we propose a hybrid algorithm of particle swarm optimization with variable local search (PSO-VLS) to solve the sub-problem of MA positioning. Numerical results validate the superiority of exploiting MAs over the fixed-position antennas (FPAs) for enhancing the SCR performance of WP-MEC systems.