Signal Temporal Logic (STL) is a rigorous specification language that allows one to express various spatio-temporal requirements and preferences. Its semantics (called robustness) allows quantifying to what extent are the STL specifications met. In this work, we focus on enabling STL constraints and preferences in the Real-Time Rapidly Exploring Random Tree (RT-RRT*) motion planning algorithm in an environment with dynamic obstacles. We propose a cost function that guides the algorithm towards the asymptotically most robust solution, i.e. a plan that maximally adheres to the STL specification. In experiments, we applied our method to a social navigation case, where the STL specification captures spatio-temporal preferences on how a mobile robot should avoid an incoming human in a shared space. Our results show that our approach leads to plans adhering to the STL specification, while ensuring efficient cost computation.