TfL have an official answer, which we'll get to shortly.
But tube stations are a surprisingly difficult thing to count, and the precise total depends on the 'correct' answers to the following questions.
Click to make your choice, and you can come back later and see if you were right.
Are King's Cross and St Pancras one station, or two?
Are Bank and Monument one station, or two?
Are Edgware Road (Bakerloo) and Edgware Road (Circle, District & H&C) one station, or two?
Are Charing Cross (Bakerloo) and Charing Cross (Northern) one station, or two?
Are Hammersmith (District and Piccadilly) and Hammersmith (Circle & H&C) one station, or two?
Are Paddington (Bakerloo, Circle and District) and Paddington (Circle & H&C) one station, or two?
Made your choices? OK, now some answers...
Above ground King's Cross and St Pancras are two very separate mainline termini, built by different rail companies on either side of a road. Before 1933 there were two separate Underground stations - one called King's Cross & St Pancras (on the Metropolitan and Circle) and one called King's Cross for St Pancras (on the Piccadilly and Northern) - at which point a proper connecting subway was built and the combination became King's Cross St Pancras. There have been several subterranean shufflings since, and the resulting complex can feel like separate stations when you're traipsing around. But for tube purposes, absolutely, King's Cross St Pancras is one station.
Bank and Monument are an interesting pair. They were two separate stations until September 1933 when the famous escalator link was opened, joining one end of the Northern line at Bank to the sub-surface lines at Monument. From this point onwards they've been staffed collectively, and when TfL count passenger statistics they're always treated as one and the same. But for station-counting purposes they remain separate, Bank and Monument are two stations. I'll prove this later.
Having twin stations on the tube map called Edgware Road isn't ideal. One's been around since the Underground opened in 1863, and the other arrived on the Bakerloo line in 1907, run by a company keen to call it exactly the same thing. Neither station has ever changed its name to avoid undue ambiguity, despite several suggestions over the years that this might be a good idea. They remain unconnected, 150 metres apart, and now on either side of the Marylebone Flyover. And that's why Edgware Road and Edgware Road are incontrovertibly two stations.
The current Charing Cross station has a very chequered history, and until 1979 was two separate stations. Previously there was a Bakerloo line station in Trafalgar Square called Trafalgar Square, and a Northern line station outside Charing Cross called Strand. When the Jubilee line was opened its new terminus connected the two, creating a portmanteau station called Charing Cross. Even though the Jubilee platforms closed 20 years later, and the other two lines remain just as far apart as they ever were, no uncoupling has taken place. Charing Cross remains one station.
The tube map appears to show just one station at Hammersmith, but the reality is very different - a Circle/H&C terminus on one side of Hammersmith Broadway, and a District/Piccadilly station through a shopping centre on the other. The two stations have the same name but separate gatelines, so when TfL are totting up their passenger statistics they always count the two separately. And so, in the first potentially controversial outcome, I can confirm that Hammersmith is officially designated as two stations.
Finally to Paddington. From the earliest days of the underground there were two stations here - Paddington (Bishop's Road) opened 1863 and Paddington (Praed Street) opened 1864. They're at opposite ends of the mainline station, and not connected underground, so were long considered two completely different stations. Only in 1947 did both end up being called the same - plain old Paddington - and the interchange symbol on the tube map has evolved several times since. What the map shows today is misleadingly connected (the Bakerloo's attached to the wrong bit), and suggests that the whole complex is one single station. But this isn't the case, nothing's changed since the 1860s, and Paddington is still deemed to be two stations.
Which brings us back to the question of how many tube stations there are altogether...
If you have a lot of patience and count up all the tube stations on the tube map, it looks very much like there are 268. But Hammersmith is actually two stations, and Paddington is actually two stations, which is why the official total is 270.
I know it's 270 stations because TfL say so on their website.
And I can confirm this using Labyrinth, the excellent Art on the Underground project which placed a maze on the wall of every tube station on the network. There are 270 Labyrinths, numbered from 1 to 270 in the same order as a Tube Challenge which took place in 2009. Labyrinth number 1 is at Chesham, and Labyrinth number 270 is at Heathrow Terminal 5. Bank has Labyrinth number 142 while Monument has 96, confirming that they are indeed different stations. Key for our purposes is that there are two Labyrinths at Hammersmith (248 and 249) and two Labyrinths at Paddington (55 and 240), should you ever go looking. And if there are 270 Labyrinths altogether, then there must be 270 Underground stations, QED.
What's also interesting is wondering how that total might change over the next few years. New tube stations are due at Nine Elms and Battersea in 2020, which'll bump the total up to 272. New tube stations probably won't be opened on the Metropolitan line extension, but if they ever are that'd increase the number to 275. And what of Crossrail? It's not going to be part of the Underground, it's going to be its own separate entity, so it won't suddenly be adding another 29 tube stations. But Crossrail's platforms will be connecting Farringdon to Barbican, and Moorgate to Liverpool Street, so will these suddenly become one station rather than two? More intriguingly Crossrail might also be linking both ends of Paddington, forming one interconnected beast, so will that finally become one station, or stay as two? All that's certain is that the official number of tube stations will remain somewhat subjective, rather than in any way obvious.