We present a novel receding-horizon multi-contact motion planner for legged robots in challenging scenarios, able to plan motions such as chimney climbing, navigating very narrow passages or crossing large gaps. Our approach adds new capabilities to the state of the art, including the ability to reactively re-plan in response to new information, and planning contact locations and whole-body trajectories simultaneously, simplifying the implementation and removing the need for post-processing or complex multi-stage approaches. Our method is more resistant to local minima problems than other potential field based approaches, and our quadratic-program-based posture generator returns nodes more quickly than those of existing algorithms. Rigorous statistical analysis shows that, with short planning horizons (e.g., one step ahead), our planner is faster than the state-of-the-art across all scenarios tested (between 45% and 98% faster on average, depending on the scenario), while planning less efficient motions (requiring 5% fewer to 700% more stance changes on average). In all but one scenario (Chimney Walking), longer planning horizons (e.g., four steps ahead) extended the average planning times (between 73% faster and 400% slower than the state-of-the-art) but resulted in higher quality motion plans (between 8% more and 47% fewer stance changes than the state-of-the-art).