The height of land in Greater London varies greatly from nearly sea level to well over 200m. Here's a visualisation of that from en-gb.topographic-map.com. This displays the elevation of the land using a colourful scale, with blue the lowest and red the highest.
Precisely what colour equals what height varies as you zoom in and out. On this map at this scale the blues are generally under 20m and the orangey-reds are over 100m. A huge swathe along the Thames and up the Lea is very blue. By contrast there isn't a lot of orangey-red, and a lot of it is beyond the Greater London boundary anyway.
Railways tend to follow lower ground because contours mean expensive engineering. That'll be why Muswell Hill and Biggin Hill don't have stations but the Lea Valley is well appointed throughout. Also, almost by definition, the Underground isn't a high level thing.
Let's start with the tube because we have a definitive answer. Back in 2011, courtesy of an FoI request, TfL released a spreadsheet listing the elevations of all 270 Underground stations. I blogged about this at the time so none of this is new. The highest station on the network is Amersham at 145m above sea level, followed by Chalfont & Latimer at 122m. Both of these are in the foothills of the Chilterns, but also in Buckinghamshire so they don't count. Instead it's the third highest tube station that's London's uppermost, and it'd be fair to say there's a clue in the name...
What's odd about London's highest tube station is that it's at the bottom of a hill. The historic town of Barnet was built at the top, and when the railway arrived in 1872 it terminated as close at it efficiently could which was some way lower down. That's why access to the station is somewhat awkward, reached down a long sloping pathway from the edge of the town centre above. A second (step-free) entrance exists lower down, closer to the estates on Underhill, but the official main entrance is the upper one with its chalet-like ticket hall and coffee-dispensing kiosk.
Highest tube station entrances 111m: High Barnet 108m: Hampstead 100m: Highgate 98m: Cockfosters 88m: Oakwood
But the height of a station entrance can be deceptive. It certainly is at Hampstead and Highgate, the 2nd and 3rd highest tube station entrances, where Northern line trains actually stop 59m lower and 39m lower respectively. You really get a feeling of descending into the depths if you ever catch a train at Highgate. Also, the column I've been using in the TfL spreadsheet says 'Ground Level Outside Station' and states that it's 'indicative', so we could really do with a more objective measurement...
At High Barnet the platforms are level with the lower entrance, indeed you can walk all the way round the back of platform 1 to get to platforms 2 and 3. We know all three platforms are 98.4m above Ordnance Datum because TfL's spreadsheet has a measurement for each - it really is impressively accurate. And High Barnet's are still the highest tube platforms in London, a tad above those at the end of the line at Cockfosters where the difference between street level and platform level is much lower.
Highest tube station platforms 98m: High Barnet 94m: Cockfosters 87m: East Finchley 82m: Oakwood 78m: Finchley Central
Note how all five of these stations are in North London, in the boroughs of Barnet and Enfield, across what might be described as the Middlesex scarp. Higher land exists to the northwest around Northwood and above Stanmore but the railways dodge that and stick to the lower ground instead. The boroughs of Croydon and Bromley rise higher still, but the tube barely spreads any distance south of the river so they don't feature. To find London's true highest station we need to leave TfL behind.
Knockholt is the farthest-flung station in Bromley, a proper outlier, indeed the Kent/London boundary runs immediately alongside, You'll find it two stops after Orpington and two stops before Sevenoaks, close to a tunnel that pierces the North Downs, hence the altitude. It's badly named, being almost three miles from the village of Knockholt, but I can see why they didn't call it Pratt's Bottom instead. I was going to say it doesn't even have a bus service, but as of this week Go-Coach have introduced a new route which runs four times a day, for what that's worth. Poor Knockholt thus sits pretty much in the middle of nowhere, a halt with a half-hourly service to Charing Cross and very little reason to visit.
Highest stations 120m: Knockholt 105m: Chelsfield
...but these are estimates based on squinting at a map, not accurate values lifted from a spreadsheet. As far as I can tell there is no freely-available list of National Rail elevation data, nor any successful FoI, so the best I can do is scan the Ordnance Survey and try to unpick the contours. When I say I think these are the only London stations with platforms higher than 100m I can only hope I'm right. But if we widen our scope slightly we can smash it...
Trams are better than trains at gradients so the heights of New Addington merit a different kind of transport connection. The Tramlink terminus thus turns out to be considerably higher than Knockholt and High Barnet, indeed technically it's higher above sea level than the top pod on the London Eye, just with nowhere near as good a view. Come instead for Betfred, Boots and Greggs, and very slightly more rarefied air than any other platform in London
Highest tram platforms 152m: New Addington 147m: King Henry's Drive 143m: Coombe Lane 123m: Fieldway 98m: Gravel Hill
These are again map-based estimates, sorry, but hopefully near enough correct. And let's finish this by abandoning rails altogether and heading to the loftiest TfL asset in the capital...
That's quite a leap from the highest tram stop, over 80m higher, and about as high as a bus stop in London could get. We're on the road south out of Biggin Hill just before it tumbles down Westerham Hill into Kent. Here the 246 bus sails past the highest point in the capital, a full 245m above sea level, about which I have obviously blogged before. Not only is Westerham Heights the highest bus stop in London but the next stop, Graham Hall Coachworks, is the highest bus stop in Kent. Muse on that the next time you're standing on the westbound Jubilee line platform at Westminster which is a full 263m below the bus stop at Westerham Heights... but that's another post entirely.