I just came across platform zero at JR Kyoto station. Are there places with more exotic station numbers, such as negative numbers, fractions or complex numbers? I'm interested in places that have such numbers as a joke, and places that have it for a serious reason.
In Amsterdam Centraal platforms are quite long, thus allowing two different trains to stop at each end of the platform. On which end of the train stops is denoted by a letter, either "a" or "b" after the number of the track. This causes huge confusion among travelers. In more than one occasion I have seen people who wanted to go to Schiphol (the airport) on the train to Lelystad (the opposite direction).
Also what's quite unusual, there are 3 tracks between each pair of platforms, with the middle track (inaccessible to passengers) is also numbered.
It's not a real platform, but King's Cross station in London has a faux Platform 9¾ (9 and 3/4), in reference to the platform in the Harry Potter novels. It has a luggage trolley half embedded in the wall, to allow people to take photographs with it.
Not really unusual but different:
In France most stations number their platforms starting at 1. But some stations have chosen to use letters instead, or mix both. This includes the Lyon station, one of Paris' busiest train terminals. The concourse is divided in two parts; one has the numbers while the other has the letters.
This is reflected in the passenger waiting process: in SNCF stations, the track number is shown 20 minutes before departure, in order to avoid passengers rushing to the platform sooner as the train is still being prepared, or another arriving train is releasing its passengers on the other side of the same platform. Before the track number appears, the concourse in which to wait is mentioned, like "Hall 1" (numbers) or "Hall 2" (letters).
Another quirk in the same area is that, a few hundred meters apart, there is another, smaller terminal, the Bercy station. It is mainly used for regional trains, sleepers trains to Italy, some chartered trains and motorail services. It has a few tracks only (let's say about 6, I do not remember the exact count), which are lettered in alphabetic order starting where the Lyon station's lettering has stopped.
For instance, the last letter used at Lyon station is N. At Bercy station, there is a track lettered R and no A.
In Dordrecht (the Netherlands), they've added tracks without renumbering the existing tracks. In particular, they've added them "before" track 1, so they logically would have to be 0 and -1. But to make it confusing, they're numbered 15, 16 and 20 (where 15 and 20 are east and west of the station building).
Platform 0 is actually quite common in Britain.
Almost all stations in Britain are numbered on the same conventional pattern: sequentially from one side to the other, counting only platforms and not platformless through tracks. Thus, when we want to add a platform (quite regular given the high intensity of timetabling strategy in this country) on the high-numbered side of the station, we number it one more than the highest-numbered existing platform, whereas if the new platform is to be on the low-numbered side of the station, such as at London King's Cross, Redhill or Stockport, we usually call it . Some places add a letter suffix instead, such as at Bedford main station, where it is called [1a] (although letter suffixes are, strictly speaking, supposed to be used for sectors of platforms, not for whole distinct platforms).
Differences in the direction the platforms are numbered is about as diverse as it tends to get in this country. Most stations either have the platforms numbered from the side closest to the entrance to the station (more predictable for passengers), or from the side which the Up (towards London) trains serve (which has the advantage that the platforms match up with other stations on the line).
The Great Western Railway numbered all of its stations from left to right when viewed from London Paddington. Most of the major towns en-route are on the south side, so this lines up with the station layout, but at those on the north side, such as in the Kennet Valley, this can mean that  is both the platform furthest from the entrance and the platform for Down (away from London) trains, and that most suburban stations in London do not have a platform 1, since the southernmost track at those places is an express track for fast trains that would not stop at a suburban station anyway. It also once led to the madness where at Bristol Temple Meads,  was both the platform furthest away from the entrance and in a new* extension. *relatively new; neuve, as opposed to nouvelle. This has since been renumbered so that the platforms are, out of step with the GWR standard, numbered from the entrance side, but in such a way that most of the platforms have 2 numbers, an even & an odd number, since they're very long & could in theory accommodate two trains. And then you have Swindon, where the platforms are numbered from north to south, out of sync with the rest of the GWR & from the side furthest away from the entrance.
Sometimes one can find such oddities as letters being used in place of numbers: Crossrail is expected to use them to maintain consistency without meddling in the numbering of the existing stations that it serves, & London Waterloo has four platforms numbered [A]-[D] to signify that they're at the end of a long corridor from the main concourse. But overall, our platform numbers vary little: they're always from one side of the station to the other, without counting platformless tracks. They also tend to each serve a particular role, such as Up fast, or Down slow, or the bay (terminal) platform for a certain branch line. On the continent, things are different. For a start, unlike in Britain, numbers usually count through tracks that have no platform as well as tracks with platforms.
The countries of the former Austro-Hungarian Empire, as well as other places, have a lot of stations where, instead of each having a platform on one side, the tracks fan out into a yard. This gives greater flexibility on a single-track line, as the stations always have more capacity than the lines (usually at least 3 tracks), so the signalman is unlikely to have to worry about there being enough capacity at the other end of the line a train is being sent down. Austria itself has mostly converted these to the western design of station, mostly to improve speed, since under the old design, trains have to slow down considerably when running through stations due to the risk of hitting a pedestrian, but they can still be found in the other lands of the empire. In these stations, the platforms are always numbered from the side where the station-house is, both because on a single-track line, there is no need to worry about consistency with neighbouring stations, & because rather than ending cleanly on the far side, the tracks usually continue into a goods yard and/or sidings and those tracks too have to be numbered.
In Germany, large stations tend to be similar to British stations in physical construction, with platforms on one side or the other of tracks, but operationally similar to Austrian stations, with the platform capacity exceeding that of the lines around, & platforms being at the signal box's disposal rather than each doing a particular job. They are pretty much always numbered sequentially from the side closest to the entrance, counting both platformed & platformless tracks, but sometimes oddities do arise, & Germany & Switzerland in particular seem to like to give platforms that don't fit into the layout randomly high numbers or even decimals. Lausanne station has through platforms numbered -, a platformless track without a number, & an east-facing bay platform, which is numbered . Stolberg Main Line, near Aachen, has 4 platforms. These are numbered , , , , . At certain other stations I have seen prefix systems & even decimals used for platform numbers ([9¾] would not be so out of place here).
When I travel to the continent, one of the first stations I usually pass through, fresh off the night ferry, is Rouen Rive-Droite. This station has its platforms numbered , , , , , , , . It took me a while to get used to it at first, but this is actually a surprisingly popular & logical way of numbering platforms. It is, for instance, used by Metro-North in the US state of New York. Bratislava's main station has only one odd-numbered platform. Its platforms are numbered , , , , , , , , . This is because it is numbered in such a pattern, but with the centre-line of the first pair of tracks being considered the centre-line of the whole railway. This numbering scheme has a multitude of advantages: the platform numbers can co-ordinate with all stations on the line, even if the neighbouring stations have only 2 platforms, as there will be  on one side of the centre-line, &  on the other; all the platforms going one way are even-numbered, while all those going the other way are odd-numbered; & because it starts from the middle, a new platform can be added on either side of the station without giving it a misfit number or messing up the numbering: simply use the next even or odd number, depending which side of the station the platform is on.
But for all this logic in France, there are also the places where the platforms are numbered in chronological order of when they were built, or something else entirely. On the same line as Rouen, we have Épône Mézières, with its , [A], [B], . There is yet Pontoise, which has 10 platforms numbered - (they have just added 10 to every platform number for no apparent reason), or Mitry-Claye's , , , ; Juvisy tries to combine one outward system with another sequential system in the same station by using a prefix, & gets , , , , , , , , , , , ; Corbeil-Essonnes takes a break from the numbered outward evens-odds pattern to become sequentially lettered for a bit, giving , [A], [B], [C], , , ; Saint-Quentin-en-Yvelines repeats platform numbers, getting , , [1bis], [B], [A], [2bis]; nearby Viroflay-Rive-Gauche repeats platform numbers more than once, for , [1bis], , [2in], [1in], [2bis]; & perhaps the cherry on the cake is Villeneuve-Saint-Georges, with its , , [1M], [1bis], [A], [B], [2bis]; or even, further out on the same line, Melun, which gives , [1bis], [A], [2bis], , [1C], [2C].
Somewhat unusual: Augsburg central station (Hbf) has the usual platforms numbered 1 through 12. Platform 1 is adjacent to the station building and 2/3, 4/5, 6/7, 8/9 and 10/12 are island platforms; all of these are through tracks. But there are some dead-end tracks that hit the platforms in the middle from both sides which can be used as platforms for short trains. These are 101, 301, 501 and 701 on the southern end and 201, 401, (I assume 601 but I would have to double-check) and 801 on the northern end.
Until recently, the station only had four island platforms, i.e. platform 8/9 was the last one so it's additionally unusual that three-digit numbers were chosen for the in-between tracks rather than two-digit ones – but maybe they already anticipated the expansion.
In a somewhat similar manner, Ulm central station (Hbf) has platforms 1, 2/3, 4/6 with a dead-end track 5, 7/8 and then additionally the tracks 25 through 29 as dead-end tracks on its southern end next to platform 1. I have absolutely no clue why they decided to start with 25 here rather than with a round number.
Most stations in the UK have platform 1 at one side of the station and then increasing numbers to the other side, letter suffixes are usually used on Longer platforms to identify parts of the platform or bays cut into the platform. But I can think of a fair few exceptions to this rule. I suspect there are many more at stations i'm not familiar with.
Platform 0 for an extra platform at the low-numbered end of the station:
Cardiff Central, Doncaster, Haymarket, Gravesend, Leeds, London Kings Cross, Rainham, Redhill, Stockport.
Platforms with only letters not numbers:
London St pancras (low-level platforms),
Liverpool Lime street, the low-level (Mersyrail) platform is referred to inconsistently. The signage at the station doesn't give it a letter or number at all. National rail live departures reffers to it as platform A, realtimetrains refers to it as platform A for trains that have not yet passed the staiont, but platform L for trains that have previously passed the station.
Platform numbering order makes no sense whatsoever:
Different numbers for different ends of the same platform:
In Saint Petersburg's Moscow Railway Station, platforms are numbered 1, 2, 3... whereas tracks are numbered 1R 1L (right, left), 2R 2L, etc, etc.
So your Sapsan train will be on track 7R.
Other stations in Russia just use separate numbering for platform and track so you can have track 4 on platform 3.