Cologne

Cologne needs 2 days: the Cathedral (the world's second-tallest Gothic church) on one, and the old town by the Rhine on the other. May-September brings the best weather, though February's Carnival is its other big moment of the year. It's cheaper than Munich or Berlin. Its Cathedral, built over more than 600 years (started in 1248 and finished only in 1880), was the world's tallest structure for 4 years after completion, and has been a UNESCO World Heritage Site since 1996.