For history & culture, Charlottenburg Palace (Schloss Charlottenburg) delivers.
The gilded apartments, porcelain cabinet and the free baroque-and-English gardens behind it give you the Versailles-lite Prussian fix Berlin otherwise lacks. The catch: it's out west, away from the central cluster, so it costs you a half-day and a U-Bahn trek for what is, by European-palace standards, mid-tier.