The emerging concept of data storytelling (DS) suggests that enhancing visualisations with annotations and narratives can make complex data more insightful than conventional visualisations. Previous works found that DS-enhanced visualisations are more effective than conventional visualisations for simple tasks like identifying key data points or the main message. However, no previous work has explored the extent to which DS enhancements influence task completion across different levels of cognitive complexity. We address this gap by presenting the results of a study where 128 participants completed tasks based on four visualisations (two line charts and two choropleth maps, either with or without DS elements) spanning a range of complexity based on Bloom’s taxonomy, which has been applied in data visualisation to categorise tasks hierarchically from lower to higher-order thinking. Results suggest that while DS-enhanced visualisations effectively support lower-order tasks (finding data points and understanding insights), they don’t necessarily aid the correct completion of higher-order tasks (application, analysis, evaluation and creation). However, DS enhancements improve how efficiently participants complete complex tasks.