In our everyday lives we regularly engage in complex, personalized, and adaptive interactions with our peers. To recreate the same kind of rich, human-like interactions, a social robot should be aware of our needs and affective states and continuously adapt its behavior to them. Our proposed solution is to have the robot learn how to select the behaviors that would maximize the pleasantness of the interaction for its peers. To make the robot autonomous in its decision making, this process could be guided by an internal motivation system. We wish to investigate how an adaptive robotic framework of this kind would function and personalize to different users. We also wish to explore whether the adaptability and personalization would bring any additional richness to the human-robot interaction (HRI), or whether it would instead bring uncertainty and unpredictability that would not be accepted by the robot's human peers. To this end, we designed a socially adaptive framework for the humanoid robot iCub. As a result, the robot perceives and reuses the affective and interactive signals from the person as input for the adaptation based on internal social motivation. We strive to investigate the value of the generated adaptation in our framework in the context of HRI. In particular, we compare how users will experience interaction with an adaptive versus a non-adaptive social robot. To address these questions, we propose a comparative interaction study with iCub whereby users act as the robot's caretaker, and iCub's social adaptation is guided by an internal comfort level that varies with the stimuli that iCub receives from its caretaker. We investigate and compare how iCub's internal dynamics would be perceived by people, both in a condition when iCub does not personalize its behavior to the person, and in a condition where it is instead adaptive. Finally, we establish the potential benefits that an adaptive framework could bring to the context of repeated interactions with a humanoid robot.