The archaeology keeps getting older. Excavations in Mitte in 2012 turned up wooden houses dated to 1174 and a wooden rod from around 1192; Spandau appears in writing in 1197, Köpenick in 1209. Cölln, the town that occupied what is now Museum Island, was first recorded in 1237, which is why Berlin threw itself a 750th birthday party in 1987.

It nearly did not survive its adolescence. Roughly half the population died as a consequence of the Thirty Years' War, and the Hohenzollern answer to an empty city was to import one. The Great Elector, Frederick William, opened Brandenburg-Prussia to religious refugees — Huguenots expelled from France, Jews expelled from Austria — and the policy stuck. Berlin became the capital of Prussia in 1701, and in 1710 several separate towns were merged into it — which is why the city still has no single centre but a scatter of them.

When the German Empire was declared in 1871, Berlin became its capital and grew at a rate that alarmed people. The population went from one million in 1877 to 1.9 million by 1900, and after the 1920 annexations that created Greater Berlin it reached four million — the third largest city on Earth after New York and London, and second only to Los Angeles in area.

The Weimar years were the peak of both its importance and its reputation. Nearly every politician, scientist and painter of the period lived and worked here. The theatres, cabarets and cinemas produced a cultural frenzy mythologised ever since, alongside severe inequality and a housing shortage the 1929 crash made brutal. In 1933 it stopped: artists and intellectuals fled, were imprisoned or were silenced, and the relatively open gay scene that had run for years despite the law was crushed.

Berlin ended the war as rubble and was divided into four occupation sectors. When the Soviets cut the land routes in 1948, the Allies supplied the western sectors entirely by air. Pilots dropped small packets of sweets on makeshift parachutes, which is how the aircraft got the name Rosinenbomber — raisin bombers — but that cargo was a rounding error. By tonnage the airlift was mostly coal, the sectors having been cut off from eastern power lines; the Allies also flew in an entire power plant and then the fuel to run it.

The West Berlin that followed was a strange political object. It never legally belonged to West Germany, sent non-voting delegates to the Bundestag, and exempted residents from conscription, which made it a magnet for students, radicals and young men avoiding the draft. The revolts of 1967 and 1968 largely happened here, ignited when police shot Benno Ohnesorg at a protest against the visiting Shah of Iran.

Berlin today holds about 3.7 million people and almost no industry. What survived the war either left the western half or was nationalised in the east and went bankrupt around 1990. The result is a capital slightly less well off than its own country's average — close to unique in Europe — and for decades that made Berlin cheap. Cheap rent bought space, and space bought experiment: studios, start-ups, clubs in buildings nobody had a use for. A former mayor summarised it as arm aber sexy, poor but sexy, and the phrase has proved impossible to shake.

The bargain is closing. Rents are catching up, and the rent question now dominates Berlin politics: a state rent-control law struck down in the courts, a campaign to expropriate the largest private landlords. Meanwhile the city keeps refilling from elsewhere, as it always has — a Jewish community growing again with arrivals from the former Soviet Union and young Israelis, tens of thousands of Syrians who came in the 2010s, Turkish and Arab Neukölln. A born-and-raised Berliner is rare.

The everyday version of all this history is cheerfully unpretentious. Currywurst — a sausage under ketchup and curry powder — costs a few euros at stands like Konnopke's Imbiss under the elevated tracks at Eberswalder Strasse, or Curry 36 opposite Mehringdamm in Kreuzberg. Berliners are famous among Germans for a directness visitors sometimes mistake for rudeness, and for complaining about their own city's administration at length — an activity guests are not invited to join.

On summer weekends the Tiergarten fills with barbecue smoke, the Mauerpark holds its Bearpit Karaoke every second Sunday, and half the city goes swimming at Wannsee. The clubs — around two hundred, techno still dominant — open late and some do not close until Monday. And the border is still legible if you know where to look: cobblestones set into the road where the Wall ran, the absence of trams in the western districts, and, at dusk, the colour of the street lights changing as you cross a line that no longer exists.