Understanding user enjoyment is crucial in human-robot interaction (HRI), as it can impact interaction quality and influence user acceptance and long-term engagement with robots, particularly in the context of conversations with social robots. However, current assessment methods rely solely on self-reported questionnaires, failing to capture interaction dynamics. This work introduces the Human-Robot Interaction Conversational User Enjoyment Scale (HRI CUES), a novel scale for assessing user enjoyment from an external perspective during conversations with a robot. Developed through rigorous evaluations and discussions of three annotators with relevant expertise, the scale provides a structured framework for assessing enjoyment in each conversation exchange (turn) alongside overall interaction levels. It aims to complement self-reported enjoyment from users and holds the potential for autonomously identifying user enjoyment in real-time HRI. The scale was validated on 25 older adults' open-domain dialogue with a companion robot that was powered by a large language model for conversations, corresponding to 174 minutes of data, showing moderate to good alignment. Additionally, the study offers insights into understanding the nuances and challenges of assessing user enjoyment in robot interactions, and provides guidelines on applying the scale to other domains.