Jumping is much, much more useful in video games than in real life. In video games it's often instrumental from saving the world from some sort of grotesque otherworldly force, not to mention getting 100%. In real life, it mostly helps keep your socks dry.
— Lore Sj÷berg, The Book of Ratings
Most human beings can't jump high or far enough for it to be a particularly useful skill in navigating platforms above abysses.
But the Platform Gamewouldn't existif we didn't relax this a bit, so the physics of jumping in video games is a bit different from that of the real world. Namely:
In real life, most humans or animals cannot jump very high. However, in a video game, it's expected that you'll be able to jump to a height at least equal to your own, if not higher.
The player character's maximum falling speed is more or less the same as their initial jumping speed, and they will land in the same state whether they fall five inches or five hundred feet. Some games may have a notion of "too far to fall", after which the fall becomes damaging, though there are usually ways of cheating the laws of physics and stopping your fall without hurting yourself.
It is possible for a jumping character to change direction in mid-air (air control), in pretty much all 2D platformers and FPSes (but not so much in third person — perhaps because being able to see yourself do this makes it clear how silly it is.) Some first person shooters even let players customize how much air control they have. In real life, most terrestrial species simply can't exert enough force in midair to effect even the slightest change to their momentum, making inertia a cruel mistressindeed. If the real-life behavior is enforced in a video gamenote the early Castlevania titles are a good example of this, characters may start feeling as if they're on an Invisible Grid.
Combining air control with accelerational falling physics makes it generally not as fast to simply run off the edge of a platform and fall straight down than to jump just short of the platform's end and fall past it. This trick has been exploited in many Speed Runs of Mega Man games.
Some characters can perform a Double Jump in midair to extend their jump's height or length, or otherwise alter its trajectory. This form of air control is increasingly attributed to magic powers, jump jets, wings, or other obvious means.
In traditional 2D platformers, the player's arms are useless when jumping: if his feet do not reach the platform, he can not grab onto it and pull himself up. Doing so is called mantling, which has become an extremely common ability in 3D games (mostly to compensate for more complex jumping controls).
Another common ability is the Wall Jump, where a character kicks diagonally against a near wall to get a further upward boost. Commonly used to ascend narrow shafts by bouncing off both sides. Though possible in real life, you're unlikely to get higher than you were before without specially designed shoes and a high-friction wall surface, and you can forget about pulling off more than one in a row without the ability to turn in midair.
Similarly, some characters can Crouch or even become Prone whilst in midair, especially common in FPSes. Which is a point used by players to access areas normally too high for a normal jump. Used perhaps when the Rocket Jump is not necessary or does not work in the present game mechanics. But can alternatively be coupled together to reach extremely high areas. This might be a way to represent the character's ability to vault over obstacles or grab ledges, which is usually very difficult to implement in a FPS.
The player should be able to jump different heights, but they should also jump as soon as the jump button is pressed (otherwise avoiding enemies becomes frustrating). Since pressure sensitive buttons were not around for the early platformers, one way of dealing with this is to have the player jump off the ground at a relatively low speed, but for the first fraction of a second continue accelerating upwards (while in the air) if the jump button is pressed. This gives the effect that a quickly tapped button gives a small jump while a held button gives a large jump. Another is to cause the player to continue rising up to their maximum jump height as long as the jump button is pressed, but to begin falling as soon as it is released.
Flutter jumps may be used to prolong jumps by floating a short distance vertically and horizontally. This move is most notably used by Yoshi in Yoshi's Island, where it has become a key move to the game, and many levels require its use.
If on a moving platform, if you jump directly upward, the platform will move on without you and you'll fall. In reality, your inertia would keep you moving with the platform (though air resistance may slow you down a bit) and you'd land on it.
If the character is lucky enough to possess a Double Jump, Rocket Boots, air dash or any other special abilities for maneuvering in midair, they are almost guaranteed to be strictly limited during a single jump but recharge instantly when you touch the ground—even if you're jumping down a pit several stories deep.
A notable exception is the 3D installments of The Legend of Zelda series, where Link's jumping ability is generally limited and "realistic". In the 2D installments, Link usually either can't jump at all, or can only jump via a magical item, in which case he has Jump Physics via A Wizard Did It. The only real exception is Zelda II: The Adventure of Link, which was radically different from the rest of the series in many other respects as well.
Enter The Matrix is an exception to the eighth rule: the character can grab ledges, and will automatically do so when falling.
The Jump Physics in Enter The Matrix are all but thrown aside when leaping in Bullet Time. It's very possible to jump around corners while in Bullet time and it keeps getting worse: You are able to attempt a jump, realise you're not going to make it and turn around to land back on the ledge you originally came from. Just shows what a freed mind can do.
One of the biggest reasons for the original Castlevania's legendary Nintendo Hard difficulty is that Belmont does not have fine jump control. Rather, his jumps are "committed", like those in real life — once he jumps, he can change his mind, but not his direction, and he always jumps the same height. This remained true through the NES games and was softened only slightly for the SNES and Turbo Duo games. However, all the 2D games since Castlevania: Symphony of the Night have allowed the player full jump control and/or Double Jump capabilities like any other Platform Game. In addition, the recent games also come with a power-up that enables the player character to jump very high, straight up into the air—and they can do this consecutively without ever touching solid ground. This grants one the amusing ability to crash into the ceiling, complete with (in some games) an appropriate collision sound and bits of falling rubble.
Jump control was limited after Castlevania: Circle of the Moon. You can change direction, but this reduces your speed. You can't change direction at all when you're swinging your weapon (though this was to bring back the "turn around and Whip It Good" maneuver from the original, which you couldn't do in Symphony because you would turn back and bump straight into the Goddamn Bats you were trying to kill).
Averted in Uncharted. While there is still certainly some creative liberties taken with realism, Nathan Drake's climbing skills don't seem all that ludicrous. The distance between jumps is usually very small when going from one wall to another behind Nathan or to the side, especially if its also higher up. From the ground, it's not possible to grab onto ledges that are more than a few inches higher than Nathan's reach with his arms straight up, and jumps made while climbing never achieve any vertical height beyond just enough to reach the next ledge, if any. Nathan's own reactions during climbing sections lampshade this; he seems to know what he's doing but is clearly worried the entire time that something could go wrong, and relieved when he makes a jump from ledge to ledge without pancaking on the ground (oftentimes far) below.
Mirror's Edge naturally does its best to avert this, as that's part of the point of the experimental game. Air control is nonexistent beyond coiling (lifting your legs in midair), height and distance are entirely based on momentum and within reasonable limits for the athletic main character, and falling damage is as realistic as the game's Walk It Off damage system allows.
Also, picking up a weapon dramatically slows the player character's speed down, and correspondingly her ability to jump high or far.
The ninja rabbit combat game Overgrowth averts most of these. You have no control over your jump while you're in the air and grabbing ledges is incredibly important in most levels. You can jump extremely high, but then again you're a rabbit. However, you can wall-run and Wall Jump all day, but you're a Ninja and the Rule of Cool applies.
Also, the wallrunning and -jumping physics are notably less forgiving than in most other games. You can't just jump upwards after falling down close to a wall, you have to have at least some momentum in the jump direction to be able to jump from a wall.
Getsu Fuma Den lets you jump in mid-air after falling off a ledge. This proves useful for collecting powerups hovering over pits.
Lampshaded in the first level of Ultimate Spider-Man. When explaining his ability to double jump Spidey admits that it is physically impossible but "so is most of the stuff I do".
The player character in Low G Man got progressively floatier jumps the more powerups you collected.
In Bloodline Champions, the Vanguard and Ranid Assasin have an ability to jump rather high and far (the Ranid twice in a row). The former damages enemies around the landing point and incapacitates them for a few seconds (but the Vanguard is perfectly fine...) while the Ranid just strikes an enemy nearby the landing point with his weapons.
In The Dishwasher, the regular jumps are pretty difficult to control mid-air, much like Mario's. However you tend to glaze over this when you get the Shift Blade (or bloodmist) and start telespamming. You forget that (A) does anything beside skip through cutscenes.
In Stampede Run, you can move around while jumping just as while running. Dropping early on jumps is sometimes necessary to avoid or break obstacles, and moving while in mid-air can get you to raised areas to get stars and bonuses.
In Let's Go Find El Dorado, the wagon will not only leap great distances from mountains, but can rotate to change its direction in midair, altering how far it flies and how it lands. Wow!
Though most fighting games make superhuman leaps a normal feat, Super Smash Bros. really pushes this trope. Instead of normal health, the meter that's worn down by damage is your character's ability to use air control! The goal of the game is to knock your opponent into the Bottomless Pit below every arena or off the side or up in the air, which is difficult prior to giving them a good thrashing because most healthy characters can:
Triple, quadruple or even quintuple jump. Plus whatever flight/teleportation abilities they've got.
Air control to the point where they can describe a full circle in the air.
Specifically damage done causes characters to fly farther and faster when hit before they regain aerial control. So you're trying to damage them enough so that a good hard hit will make them fly so far that they can't make it back to safe ground or just get KOed instantly.
Virtua Fighter is known for its adherence to realism...in all things except jumping. It's very weird to see Sega retain the super-high, extra-floaty jumps after six or seven games, especially when Tekken ditched it after the second game while simultaneously making each game in the series more unrealistic than the previous one.
The Dissidia: Final Fantasy games have jump physics, though they resemble nothing like regular physics. The number of jumps you can do in a row ranges from two to sixteen, you have free control of your momentum when evading, falling, or dashing. Your dashing can be done without any shown source for such movement, and in directions directly clashing with gravity. Also you and your opponent can pause in the air to swordfight. And this is without getting into how wall running works in this game. So, in short, this is your jump physics on drugs.
First Person Shooter
While the marine you play in Doom cannot jump, he can fall from the Empire State Building and not suffer even a sprained ankle. Some source ports of the game, however, allow jumping to be enabled, as well as falling damage.
It kinda sorta makes sense in the first episode of the original Doom, where you're on Phobos. The gravity on Phobos is so weak that you probably can run as fast as a horse while carrying a dozen of guns without breaking a sweat, and fall from great height without risking injury. On the other hand, if you try to jump, you may achieve escape velocity and collide with the forcefield holding the atmosphere in place. (There has to be a forcefield. It doesn't make sense otherwise. Oh well, it doesn't make sense anyway.)
You would need all that gear weighing you down or the first step you took while trying to run would launch you into the ceiling.
All those facts go flying out the window in Doom 2, which has the exact same marine, with the exact same equipment plus one gun, running at the exact same speed, on Earth.
In the Half-Life series, the player is often required to jump, then hold the duck button to hold their legs up, allowing the player to reach slightly-higher ledges or clear slightly-longer chasms. This proved so useful that every Valve game since has made use of it, confusing players who are not familiar with the mechanic and confounding players who go on to play other games.
Air Control is also featured, especially blatantly in Xen of Half-Life, but you do have to collect a special power-up to let you do egregious physics violations.
Inertia was kind of fixed in Half-Life 2, at least when moving on trains. However there's a section where you jump on a speeding train, and inertia no longer applies.
Star Wars: Dark Forces II: Jedi Knight does the hold-for-longer-jump thing, but you don't actually jump until you let go of the button. The game for Spider-Man 2 does pretty much the same thing.
Force Jump in the original Jedi Knight is also egregious: it's possible to a) kill yourself by jumping into a low ceiling with full force jump power, and b) kill yourself by using force jump to jump as high as possible, then take falling damage by landing back on the exact same spot you jumped from.
Most of the Medal of Honor and Call of Duty series of games the player can jump half the height of the door, and turn in midair, as well as crouch. The fact that none of that affects one's mobility is a point exploited by players to make being shot at that much more difficult. Later games made jumping begin to slow down players for a short time (Repeatedly jumping making the character almost at crawl speed in later Modern Warfare games).
In Crysis, the strength mode on your suit allows you to jump much higher than usual, allowing you to scale cliffs and jump on top of buildings.
A lot of early 3D games with more primitive jump physics don't actually have jump physics beyond "what goes up must come down." This is especially noticeable in cases like Rise of the Triad and Super Mario Kart, where the character could move/steer just as easily in the air as they could on the ground.
Halo; in the first game it was kind of justified, considering you were a superhuman in a possibly low gravity setting (although it is stated to be otherwise). Halo 2 and Halo 3 are set on Earth, where the height at least is justified but he still falls too slowly. Halo 3: ODST is set on Earth, and the PCs are a squad of normal human being with no visible jet packs. Now you can't jump as high, but your jump is still pretty floaty.
Also, in the first game, Master Chief died rather easily from fall damage. In the 2nd and 3rd installations, fall damage was removed (handwaved by his new suit being able to withstand the falls).
The first Quake allowed the player to change directions and move freely while in the air (called "air control" in the Quake fandom), but this was taken out in Quake II and later games.
Quake's approach to conservation of momentum was a bit odd, with the end result that you could jump higher when running up a hill, due to the already existing upward momentum. That was fixed eventually.
The strange quirks in the way jumping and momentum interacted in Quake created one of the earlier instances of bunny-hopping, where serial jumps would get progressively longer and faster, until your speed would put racehorses to shame.
Urban Terror, a game based on Quake III, has special maps (jumpmaps) where you go through an obstacle course using the various jump physics. As it has wall-jumping, ledge-grabbing, air control, circle-jumping (turning the mouse while in the air makes you go faster), bunny-hopping and powersliding (crouching in mid-air makes you slide on landing), these courses can get very creative.
The Unreal series increased the player's air control in many of the successive sequels. Double-jumping was added in UT2003/UT2004, with an optional setting to allow quadruple-jumping "when double-jump just isn't enough."
The Scout from Team Fortress 2 is the only class that can double jump. He can also utilize an aluminum baseball bat named "The Atomizer" for a third jump at the cost of 10 health), use the Recoil Boost from an incredibly Sawed-Off Shotgun known as the "Force-A-Nature" to give himself another "jump" on top of that, or hit himself with the Boston Basher (a bloody bat with spikes sticking out of it) to reset his jumps in mid-air, at the cost of a lot of health. And there's the rocket/grenade jumping the Soldier and Demoman use...
Jumping in Redneck Rampage is, for one single reason, just plain weird. Instead of Leonard's leaps being in a parabolic arc like you'd expect, his trajectory is best resumed as two lines forming an angle: he jumps and goes up at a constant speed until he reaches a certain height, then immediately he falls just as fast. Needless to say, it makes platforming in the game very frustrating.
Normal jumping in City of Heroes follows fairly realistic physics, with next to no air control. However, even the first power from the Leaping pool lets a character jump from a platform and land on the next level up, dodging the ceiling and banister on the way there.
It does, however, have a sliding scale for falling damage, so the greater you fall the more damage you take. You can't die from a fall, however, only be reduced to a single hit point. Which still can be rather humiliating for your big bad buffed out invulnerable tank when she then gets taken out by a lowly minion who managed to get a hit which you normally wouldn't even notice.
That said, a Hero with no jumping related powers can leap a good six to ten feet vertically. Conversely even a hero with several stacking super-jumping powers can only leap modestly tall buildings. Skyscrapers are right out.
Averted in the MMORPG Vanguard, where jumping is considerably more realistic as direction and lateral speed are fixed during the jump. Compare to the MMORPG EverQuest II where both speed and direction can be changed at will mid-jump (which leads to arc-jumping around obstacles, among other results).
One cannot grab ledges in World of Warcraft; many players have fallen to their doom after missing a jump by inches. However, platforming is rare in the game and mostly limited to dungeons.
FusionFall contains jumping platforms in dungeon-like areas called Infected Zones, the platforms in Vilgax's ship contains platforms created by Gwen Tennyson, whom had pink platforms that looks very different compared to Infected Zones platforms.
Guild Wars 2 jumping allows small mid-air turns. The distance of your jump is determined by how long you hold the jump key. Your forward motion slows down a lot when you release the key causing some very odd jump trajectories. How far you can jump seems to vary a lot in jumping puzzles. Many of the jumping puzzles would be quite simple if the character could climb up onto a chest-high platform. You stop when you land on a surface, even if it is icy or round. While the distance you can jump is not as absurd as in many games, often it is possible to jump up 1.2 meters from standing.
Dungeons & Dragons Online uses a player's Jump skill to determine how high and far they can jump. Unless a player invests a lot of points into it, the distances are realistic. However, the player can change direction and speed in mid-air as well. Besides that, a number of spells effect how players jump and fall, like Feather Fall and Head in the Clouds.
Note that originally in Donkey Kong, Mario couldn't change direction in midair and falling beyond his jump height would kill him. Mario Bros. would give him and Luigi the ability to survive falls (necessary as jumping between levels was part of the gameplay) and in Super Mario Bros they had just enough midair control to cut short a forward jump.
The primary difference between Mario, Luigi, and Peach was in their jumps: Luigi flutter-jumps, and Peach can hover.
The Mini Mushroom in New Super Mario Bros. changed Mario's mass, but not his jump power, allowing for comparatively super-long jumps.
The 3D Mario titles included fall damage past a certain height, but you could Ground Pound to reset your momentum, trading air control for protection from the fall. This mechanic was scrapped since Super Mario Galaxy, reverting Mario's landing ability to that of the 2D games. Though doing that now causes Mario to crash and unable to move for a while.
One of the major sources of difficulty in New Super Mario Bros U stems from the fusion of 2D platformer level design with Jump Physics similar to those in the 3D games, especially in how Mario's inertia is conserved while jumping
In Mega Man, characters can be steered while falling, and Bass can double jump in Mega Man & Bass.
The Mega Man X, Zero, and ZX games gave the player a useful wall kick, as well as air-dashes under certain conditions. The wall kick can be combined with air control to climb up vertical walls by jumping off them repeatedly.
In the Commander Keen games, the protagonist has a pogo stick, which enables him to make very high jumps. In Commander Keen 4, the protagonist also gains the ability to grab onto the edge of walls and pull himself up.
Another exception: the 2D Prince of Persia games, known for their realism at the time; the animations were actually hand-rotoscoped from live footage. The hero's movements have more to do with real humans than most computer sprites. (He can still jump maximum distance, slam into a wall, fall ten feet, grab onto a stone projection by his fingertips, chin himself, and then do it all over again.) Newer games are still fairly realistic, but with more Le Parkour.
N combines extremely liberal jump physics with regular fall damage. Get too creative with your flying around and you will splatter. If you go too fast in wall jumps, you can splatter on the ceiling.
An interesting aspect of N is that whether or not a fall kills you depends on your velocity perpendicular to the surface you strike. For example, you can fall any distance straight down and survive if you land on a slope of more than ~45 degrees.
Rayman has an interesting time with the eighth rule, making ledge-grabbing one of the skills the main character actually has to learn.
A more gentle use in Ghosts N Goblins. Arthur's jump is not particularly high compared to that of other platform game characters, and its direction can't be changed once started.
In Super Ghosts 'n Goblins (and possibly other games as well), Arthur does have a double jump and can choose to aim in a different direction than his first jump, though he will then be committed to the new trajectory. Also, he can't use a mid-air jump if he walks off of a ledge (which also causes him to fall straight down even if he was running... Selective Inertia?)
Bionic Commando offers a unique approach to jump physics: You can't jump at all, and instead you have to rely on your bionic arm to get around. The developers of the remake, Bionic Commando Rearmed noted that the swinging physics are in complete defiance of the laws of motion, but the Rule of Fun keeps it all valid.
Crash from Crash Bandicoot has three types of jump: normal, double and crouch jump.
The Metroid series plays with this trope every way to Sunday. Depending on which game you're playing and which power-ups you've collected, you may be able to jump, high jump, bomb jump (which can almost always be chained at least once, sometimes indefinitely; it is also possible to bomb jump diagonally up and to a side, which can of course be chained, sometimes indefinitely), wall jump, double jump, space (infinite) jump, grab ledges, and/or kill enemies by jumping into them.
Jump physics has become such a trademark of the franchise, in fact, that when Nintendo first announced the Metroid Prime series, many fans' first reaction was "How the hell are they going to do jumps in first person?"
I Wanna Be the Guy usually plays this trope straight (with a double jump, a Wall Jump in a few cases, etc.) but in the level where you're on a cart, your inertia is actually maintained and if you simply jump without trying to go in either direction, you'll land right back on the moving cart.
Notable is LittleBigPlanet for actually conserving momentum when jumping; if you were on a moving platform when you jumped, you would still be moving along with it (though air friction might move you back if the platform was fast enough), and jumping on an upward-moving platform provided a noticeably higher jump, again depending on the speed.
In Super Mario Galaxy, if you jump on a moving platform, you actually don't stay in place. All planetoids exert a force of 1 G, regardless of their size.
Also don't forget the fact in his playthrough we find out that Luigi apparently butters the soles of his shoes wherever he goes (although he's been like that on and off since Super Mario Bros 2, US and Japan).
Croc, you conserve momentum when jumping from a moving platform. It goes further than that, simply being above a moving platform gives you its momentum (angular or not), even if you didn't jump from it,
In Spelunky, your character is one block tall, and can jump up only one block. He can grab a ledge and mantle up (or down to hang from it). Falling damage is brutal but falling into water is safe. But there is a springboard effect when landing on enemies. You also jump less or more by tapping the jump button or holding it down, respectively.
In Jumper, we have Ogmo, who can Double Jump, has no moving platform inertia, push crates around while having no arms and bounce off cannonballs. Jumper Two gave him the ability to perform wall jumps, skid jumps and slippery ice which disables skid jumps.
Head Over Heels offers different mechanics to each half; Head, being the top half and therefore having the arms, and thus wings, can jump twice his own height, glide, and control himself in the air. Heels is the bottom half with the feet; this gives him Super Speed (relative to Head), but he can only jump his own height and falls like a stone as soon as he reaches the peak. The combined being gets Head's jump powers and Heels' speed.
The first Teenage Mutant Ninja Turtles game for the NES had awful jumping physics: the turtles nearly reached the top of the screen, descended slowly and while you could change the direction midair, the slightest touch in the D-Pad made the turtle would turn abruptly keeping the momentum, forcing the player to make unnecessary high jumps and tapping left-right madly to make the character land safely. It doesn't help that after area 3 it is required to make awfully precise jumps almost all the time.
In Rod Land, the ability to jump can be enabled or disabled by option. The way to get to higher platforms, however, is by placing ladders.
In Donkey Kong Country, the key to grabbing many goodies or even sometimes just to progress is to roll into the air and bounce off of nothing at all. It's how you're supposed to pick up all those bananas and coins halfway down a bottomless pit without dying.
Bomb Jack was one of the first Platform Games to give the player full air control, with controls allowing Jack to jump higher or not so high and fall slower or faster. Mighty Bomb Jack kept these jumping mechanics, but Bomb Jack II changed them completely: Jack can jump only to platforms that are directly above, below, or to the side.
In some Spyro the Dragon games, Spyro is able to double-jump, and in most games he can "hover", which is basically another name for "mantling". Justified in that he's a dragon with wings.
Athena has some bizarre jumping physics that make Athena jump higher after completing one normal jump. This happens once she gets the Shoes of Icarus, or from the start in the NES version.
Jak and Daxter's Jak must have some amazing upper body strength. If timed correctly, it's possible to jump, go into a dive to punch the ground and jump again right when Jak's fists connect, essentially pushing himself up by the arms—up to three times his body height. Falls from great height cause damage, but it's possible to avert this by doing a Spin Attack right before hitting the ground, slowing him down.
In Jetpack, you can jump (and control your falling direction) when the Jet Pack is out of fuel, but jumping is slow and gains only one block in height.
Jumping is different for every character in The Angry Video Game Nerd Adventures. The Nerd jumps normally, while Mike has the highest height. The Bullshit Man and the Guitar Guy suffer somewhat in this department, but make up for it elsewhere (the former has a Double Jump while the latter can run faster).
In Summit, you can turn in mid air, and also survive jumps from any height. The main character will take a second to recover though.
In Portal, Chell is able to survive falling a hundred feet, getting thrown across a room the size of a city block, reaching terminal velocity, etc. because she has springs attached to her ankles.
According to the commentary track, the springs were added after playtesters started asking questions. It is still not explained why she can change directions in midair.
Also, despite (or because of) the springs, Chell can't jump higher than a human realistically could, forcing you to use portals, or a handy Companion Cube to reach higher platforms.
In The Quest Of Ki, Ki can levitate to any height by holding down the jump button, but can't jump in mid-air. Also, if she hits the ceiling, she will fall straight down and get stunned; as one in-game hint says, this is actually important for getting through some spots. This is because it's the only way for Ki to duck.
Role Playing Game
In Final Fantasy VII's final dungeon, the platforms spiral downward, so the spacing isn't really an issue (though, in real life, jumping from one rock to the next would be difficult and rather painful). The problem occurs if/when the player decides to go back, and Cloud is mysteriously able to leap fantastic distances. Also used in The Movie, liberally, pushing the limits of Willing Suspension of Disbelief.
Vagrant Story features a surprising amount of platforming/Le Parkour for a... hardcore Squaresoft stat-centric RPG set in classic Ivalice. Since the main character's agility stat determines his platforming skills, some parts of the game are simply closed off until New Game+ because it's literally impossible to get your stat high enough the first time around.
Not being able to jump high enough to get past a certain crate leads to Game Breaking Bugs, if you don't have a long-range weapon to break the crate with.
Also, Ashley can change jump directions in mid-air... which starts to look just plain silly after his stat is maxed out.
The acrobatics skill in The Elder Scrolls IV: Oblivion: high levels allow characters to jump several times their height, vertically. If this wasn't enough, the highest level allows you to jump off the surface of water.
In The Elder Scrolls III: Morrowind magic scrolls that buff this stat to ridiculous proportions can be found on the smashed corpse of a wizard. These allow you to leap tall buildings in a single bound, but take care to have a second scroll ready for the landing, since the buff will usually wear off in mid-air - otherwise Fatality Ensues.
In the Hunter The Reckoning game for the GameCube, jumping committed a character to flying in a given direction until she landed. Like many other games you were also allowed to freely change your facing in mid-air, probably to allow you to jump, flip around, and fire your guns at pursuing zombies. In any multiplayer game however, players will quickly realize that they can spin wildly in the air like a top while jumping, without losing any momentum or slowing themselves down. Doing this doesn't offer any benefits but looks hilarious and so frequently becomes many players' default method of moving through already cleared areas.
Any Final Fantasy Dragoon can Jump...and stay in midair for a long time. Pretty much defeats the point of having an armored character, though. In Final Fantasy Tactics Advance, though, the jump attack, like all other attacks, is instant.
Shoot Em Up
Contra Force has a famous "pause jump" bug: if the game is paused when the character is performing a jump, then, after the game is resumed, he will still fly up the distance of a jump. And then it can be done again. That is, with perfect pause timing, one can literally FLY.
Halo Zero takes this to JUMPING THROUGH THE TOP OF THE SCREEN, at times.
Stealth Based Game
The (unfortunately) forgettable sci-fi stealth game Stolen allows and requires the standard leaping, wall-jumping and mantling, but outside of hopping in place (and as you can whistle, this is even redundant for deliberately making noise) only allows jumping to be done in specific locations; the jump key normally executes a roll. For example, while you can walk up onto a toilet to reach a wall vent, you can't jump onto the toilet. Likewise, you can't in any way move onto a desk or jump off of a "nonessential" crate. The game does thoughtfully display all jumping and climbing-related keys when they're usable, but the constraints on movement are still evident.
Krauser from Resident Evil 4 displays some crazy jumping ability. Leon can't even jump over crates stacked oddly on top of one another. Otherwise the jump physics were pretty good.
Third Person Shooter
Crackdown's cybernetic main character can vault several city blocks when his Agility skill rises high enough.
Shadows of the Empire completely averts this trope in its train level. The train sometimes turns, sharply (for a train, at least), and if you jump you're likely to land to the train's side just as you would expect. The trick comes in waiting for straight sections, or jumping lightly and from the inside.
Wide Open Sandbox
Grand Theft Auto IV has fairly realistic physics in this regard. Standing still, Niko can't get more than a few inches of height. A running jump can clear a few feet, but not more than you'd expect. And, of course, inertia takes its toll if you decide to jump on a speeding train or some such. Although the sprinting animation is a lot faster than the jumping animation, so if you jump while at full speed the difference apparently evaporates into the ether and Niko slows right down.
However, in San Andreas, the Jump Physics are all over the place. CJ himself can't jump very high (although he can abuse Le Parkour), but if you max his Biking stat, stick him on a Mountain Bike, and bunny hop, he can scale Mount Chilliad in a couple of minutes-and fall off it with no hit point loss.
Prototype: Alex Mercer can do some pretty insane things, starting with jumping thirty feet in the air and working up to airdashing back against his previous momentum. Justified as he constantly mutates new upgrades to meet demand; there's an entire upgrade category dedicate to physics-warping stunts like this.
Don't forget running horizontally along the outside of a cloudscraper, reaching the corner of it, and magically sticking to the wall at a 90 degree angle from your previous one. Of course, this is all wizardry.
One of the signature elements of the Gothic series (to the point its absence is one of the reasons 3 and 4 are considered to not exist) is the context sensitive jump system, where instead of the default standard barely controllable Real Life/Belmont long jump, the hero will automatically do a standing jump if under a ledge and try to grab it, climb over any low edges in his path or pull himself up if any ledge above him is low enough. The physics get very gamey however, if the player is trained in the acrobatics skill (5 skill points in Gothic 1 or upon reaching a given point in dexterity in Gothic 2 with Night of the Raven), which makes his jumps turn him into a fully controllable missile.
Non-video game examples:
Anime and Manga
In The Lucifer and Biscuit Hammer, Yuuhi puts a lot of time into using his domain control to imitate jump physics, gradually giving him impressive maneuverability. Several of the other Knights follow suit to fit their fighting styles. Sami can clear low mountains by the end of the series, using her powers as Princess to boost her physical abilities to superhuman levels. Lacking any kind of intervention via powers, though, realistic physics are in play.
In the first episode of Sonic X, the cops set up a barricade to stop Sonic. They are sorely disappointed when he jumps 100 meters over it with little effort. Sonic repeatedly impresses people with his jumping just as much as his speed.
A core competency of the Jedi and the Sith in Star Wars. Of course, since they are using The Force, normal laws of motion need not apply.
Chinese Wire Fu films like Crouching Tiger, Hidden Dragon, have people jumping absurd distances, floating through the air really slowly, and changing direction to the point of complete 360s.
Parodied in Kung Fu Hustle with the character of The Beast, whose "Toad Style" of fighting grants him superhuman jumping abilities, including being able to clear a whole room in one leap from a standing start and speeding into the sky like a rocket.
Blade features vampires and the title character jumping higher, farther, and faster than humans would.
Underworld has a scene of Selene jumping from the top of a building, sticking the landing in a crouch and instantly straightening up and walking away.
While vampires are tougher than humans, no explanation is made for the pavement she lands on or her shoes. Both appear to be Made of Iron.
The Matrix movies have characters frequently roof-hopping across multilane thoroughfares, though they do at least concede the need for extra height to make the distance.
In The One, both of Jet Li's main characters can jump higher than a normal human and have a much greater control over their body while in midair. The first is justified by the in-universe Conservation of Ninjutsu (the Evil!Jet Li has killed most of his duplicates in The Multiverse, and all the life force has thus been re-divided between the remaining two). The second, not so much.
In the original Starship Troopers novel, the Powered Armor of the titular Space Marines uses a combination of strength-augmented legs and proximity-activated jets to allow for really long, or high, jumps. This is also a method for getting around; they bunny-hop instead of running.
In the web-novel Domina, physics get weird when Akane uses her Super Speed. Not only can she jump higher and farther, but she can survive impacts and falls with little difficulty. No explanation has been given, though in fairness the characters are just as bewildered as the audience.
In the webcomic 8-Bit Theater, Thief once avoided injury from a huge fall from a crashing airship by simply double jumping at the last second.
There's also Dragoon, who can jump ridiculously high.....as an attack (mostly only used on Black Mage), but can't jump very high when it's not an attack.
Black Mage once meets him in midair after being launched by an explosion, and Dragoon promptly stabs him with his spear and uses him to safely land on. He then thanks Black Mage and leaves. Apparently he can start jumping like that without a specific target, but he needs one to execute his combat maneuver on in order to land without hurting himself.
Samurai Jack, in the episode where he is taught how to "jump good," after the style of the above-mentioned martial arts films.
Avatar: The Last Airbender makes heavy use of this trope in most of the fight scenes. Aang is an Airbender, so it would make sense for him to be able to jump abnormally, but other non-airbending characters can jump higher and farther than normal people.