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Eye gaze plays an essential role in the organisation of human goal-directed behaviour. Stationary gaze entropy and gaze transition entropy are two informative measures of visual scanning in different tasks. In this work, we discuss the benefits of these eye gaze entropy measures in the context of driving behaviour. In our large-scale study, participants performed driving tasks in a simulator (N = 380, 44% female, age: 20-73 years old) and in on-road urban environments (N = 241, 44% female, age: 19-74 years old). We analysed measures of eye gaze entropy in relation to driving experience and compared their dynamics between the simulator and on-road driving. The results demonstrate that, in both driving conditions, gaze transition entropy is higher, whereas stationary gaze entropy is lower, in more experienced drivers of both genders. This suggests that gaining driving experience may be accompanied by a decrease in overall gaze dispersion and an increased unpredictability of visual scanning behaviour. These results are in line with previously reported trends on experience-related dynamics of eye gaze entropy measures. We discuss our findings in the framework of the system-evolutionary theory, which explains the organisation of behaviour through the history of individual development, corresponding to the growing complexity of individual-environment interactions. Experience-related dynamics of eye gaze complexity can be a useful factor in the development of practical applications, such as driver monitoring systems and other human-machine interfaces.