We introduce and study a set of training-free methods of information-theoretic and algorithmic complexity nature applied to DNA sequences to identify their potential capabilities to determine nucleosomal binding sites. We test our measures on well-studied genomic sequences of different sizes drawn from different sources. The measures reveal the known in vivo versus in vitro predictive discrepancies and uncover their potential to pinpoint (high) nucleosome occupancy. We explore different possible signals within and beyond the nucleosome length and find that complexity indices are informative of nucleosome occupancy. We compare against the gold standard (Kaplan model) and find similar and complementary results with the main difference that our sequence complexity approach. For example, for high occupancy, complexity-based scores outperform the Kaplan model for predicting binding representing a significant advancement in predicting the highest nucleosome occupancy following a training-free approach.