I can't believe nobody has yet mentioned the book Period mappings and period domains by Carlson, Müller-Stach and Peters. Chapter 1, the introduction, is written almost as a story, starting with the pure Hodge structure on the cohomology of compact Kähler manifolds and introducing mixed Hodge structures by looking at degenerations. Further niceties await the reader in subsequent chapters.

I can't believe nobody has yet mentioned the book Period mappings and period domains by Carlson, Müller-Stach and Peters. Chapter 1, the introduction, is written almost as a story, starting with the pure Hodge structure on the cohomology of a Riemann surface and introducing mixed Hodge structures by looking at degenerations. Further niceties await the reader in subsequent chapters.