Predicting demand for bike share systems (BSSs) is critical for both the management of an existing BSS and the planning for a new BSS. While researchers have mainly focused on improving prediction accuracy and analysing demand-influencing factors, there are few studies examining the inherent randomness of stations’ observed demands and to what degree the demands at individual stations are predictable. Using Divvy bike-share one-year data from Chicago, USA, we measured demand entropy and quantified the station-level predictability. Additionally, to verify that these predictability measures could represent the performance of prediction models, we implemented two commonly used demand prediction models to compare the empirical prediction accuracy with the calculated entropy and predictability. Furthermore, we explored how city- and system-specific temporally-constant features would impact entropy and predictability to inform estimating these measures when historical demand data are unavailable. Our results show that entropy of demands across stations is polarized as some stations exhibit high uncertainty (a low entropy of 0.65) and others have almost no check-out demand uncertainty (a high entropy of around 1.0). We also validated that the entropy and predictability are a priori model-free indicators for prediction error, given a sequence of bike usage demands. Lastly, we identified that key factors contributing to station-level entropy and predictability include per capita income, spatial eccentricity, and the number of parking lots near the station. Findings from this study provide more fundamental understanding of BSS demand prediction, which can help decision makers and system operators anticipate diverse station-level prediction errors from their prediction models both for existing stations and for new ones.