Munich

Munich is the city where Germany contradicts itself best. It is the country’s richest city and its most conservative, home to imperial museums and insurance companies, and at the same time the place where people surf a wave inside a public park and can bring their own lunch to the market and just buy the beer. The old town sits inside a ring one and a half kilometres across and is entirely walkable: the square with the carillon, the brick cathedral, the Wittelsbach palace and a park bigger than Central Park all sit within twenty minutes of each other.
Two things worth knowing before you go. The first: Munich does not hide its twentieth century, but it does not flag it either — the bronze strip in the alley behind the Feldherrnhalle, the plinths of the Nazi temples at Königsplatz, the White Rose leaflets set into the pavement in front of the university are all there, with no signs, and you have to know where to look. The second: on Sundays the Bavarian state museums cost one euro, Pinakotheken included, while the Viktualienmarkt is closed: it is worth building the days around that.
In one day you can do Marienplatz, the Alter Peter tower, the market, the Frauenkirche, the Residenz and the evening in a beer hall. In three you add the Pinakotheken, Königsplatz with its two overlapping histories, the 1972 Olympiapark and Nymphenburg Palace. In five you leave the city, and that is the real reason to stay: Neuschwanstein and Ludwig II’s castles, Salzburg two hours by train away, or the Königssee, where only electric boats have sailed since 1909.