Explore a comprehensive lecture on nonadditive entropies and statistical mechanics at the edge of chaos. Delve into key concepts such as Boltzmann-Gibbs entropy, Q-logarithm, additivity, and extensivity. Examine the historical context with references to Einstein, Planck, and Gibbs. Investigate the Central Limit Theorem, Logistic Map, and Standard Map. Analyze the Second Principle of Thermodynamics and the validity and failure of the Boltzmann Weight. Discover applications in complex networks and high-energy physics. Gain insights into cornerstone principles that shape our understanding of statistical mechanics and entropy in complex systems.