Modern manipulators are acclaimed for their precision but often struggle to operate in confined spaces. This limitation has driven the development of hyper-redundant and continuum robots. While these present unique advantages, they face challenges in, for instance, weight, mechanical complexity, modeling and costs. The Minimally Actuated Serial Robot (MASR) has been proposed as a light-weight, low-cost and simpler alternative where passive joints are actuated with a Mobile Actuator (MA) moving along the arm. Yet, Inverse Kinematics (IK) and a general motion planning algorithm for the MASR have not be addressed. In this letter, we propose the MASR-RRT* motion planning algorithm specifically developed for the unique kinematics of MASR. The main component of the algorithm is a data-based model for solving the IK problem while considering minimal traverse of the MA. The model is trained solely using the forward kinematics of the MASR and does not require real data. With the model as a local-connection mechanism, MASR-RRT* minimizes a cost function expressing the action time. In a comprehensive analysis, we show that MASR-RRT* is superior in performance to the straight-forward implementation of the standard RRT*. Experiments on a real robot in different environments with obstacles validate the proposed algorithm.