"Two minutes into the game and you're already dead! Is there something in your family tree that causes dysfunctional locomotion?"
— Space Quest II, if you walk off of the space station into deep space
These are games that can be lost on the very first turn or two, or very shortly after you begin a level.
This can be deliberate through a Kaizo Trap, Schmuck Bait like a suicidal dialogue option or Press X to Die, or an "Impossible" difficulty level; and may or may not result in a Nonstandard Game Over. It can also be accidental, due to a Game-Breaking Bug, bad starting positions in Randomly Generated Levels, or simply taking an action that the designers never accounted for.
Note that it's always possible for extremely contrived stupidity to result in a player death - mundane cases like failing to properly jump or steer your vehicle around elementary obstacles don't cut it. In the same vein, unwinnable user-created content isn't worth mentioning unless the deaths are funny or interesting.
In Takeshi's Challenge, if you choose to PUNCH the man who asks for your name at the very beginning, you'll get a game over before the game itself even started. Funny, yet also quite bitchy.
Uninvited starts you off inside a car on fire. You have only a few moves to open the door and get out before the car explodes with you inside.
In The Colony, the first thing you need to do is turn on the lights on your crash-landed ship. The light switch is one of two identical unmarked buttons on a console. What does the other button do? Blows up the planet.
Karateka: At the beginning, get into a fighting stance, then stand back up. You fall off the ledge behind you. Game over.
It is theoretically possible to die before your first move, because the Random Number God left an artifact of a different alignment than your character on the starting tile, and you had autopickup turned on. The artifact will zap your character for trying to touch it, and the damage might kill your character. This is extremely unlikely though. Player ais523 demonstrated this death and calculated the odds at about 1 in 3 million.
Slouching Towards Bedlam lets you jump out the window as your first command. Depending on your point of view, this is the best ending, and definitely one of the best two.
Red Faction: Guerrilla has one where if you attack your brother in the first mission, you get a game over screen complete with "WTF, you killed your Brother!?!?!?!?!"
In Suspended, selecting the Impossible difficulty level causes the sun to go nova a few turns in, destroying the planet.
In the original Wing Commander, when you started the game, you immediately, and without warning, found yourself in the cockpit of a starfighter, with cheezy arcade-style music playing. No matter what you did, you died within seconds, only for the game to reveal that this was your character messing around in the flight simulator.
In Grand Chase, one PVP map has a layout where, if you're unlucky enough to be in a certain position, you lose a life immediately after spawning. (You still have 4 lives, but you're at a serious disadvantage.)
In King's Quest I, it's not unusual to die by falling into the moat on the first screen because you can't navigate the small, wooden bridge properly while adjusting to the controls.
Akalabeth, the spiritual precursor to the Ultima series of games, begins with your main character in a general store and enough gold to equip him. If you decide not to purchase any food - which is necessary for all movement in the game - your character drops dead from starvation instantly as soon as you leave.
Subverted in Bastion. If you fall off the path in the opening level the Lemony Narrator will say "...and then he fell to his death... only foolin'" before The Kid lands nearby and takes damage from the fall as would happen at every other point in the game.
In Elite and its Fan RemakeOolite, beginning players must dock with space stations manually until they can afford to buy a docking computer for their ship. The catch is that all orbital space stations rotate, making said docking a hair-raising experience at best the first time it is attempted and causing a number of new pilots to plow into the station instead of flying into the docking bay. Engaging pirates before being upgraded with advanced weapons, armor, scanners, or fuel injectors also tends to lead to disastrous results.
The dark colors of Another World's first screen, in which the protagonist suddenly appears in a body of water inhabited by an unseen, tentacled monster, does not make it at all clear that the best choice of action is to immediately swim upward to safety. (In fact, it doesn't make it clear that the opening cutscene has ended.) If you don't move quickly enough, you'll be killed within the first five seconds of game play.
Succeeding in that, if you don't leave the second screen immediately, the monster will reach up and snatch you back down, killing you anyway.
This can also happen in the bug-riddled Sonic the Hedgehog (2006). Protip: If you don't start a level already on the ground, don't touch anything.
King's Field The Ancient City: Walk forward for about five seconds from the start.
Action 52 had one—in StarEvil (a vertically-scrolling shoot-em-up), there's a Deadly Wall at the very start of the game. Not dodging it might seem like People Sit on Chairs—except it's so close to the start that you literally have less than one second from exiting the title screen to dodge it. The game also has Hambo's Adventure, which has randomly generating enemies, and the ability to game over less then a second.
The first Leisure Suit Larry game: Walk south on the road. You become an instant pancake. Game over. Walk into the alleyway to the west, get beaten to death by a mugger. Game over.
The first Space Quest (original) game on a modern computer: Set speed to "Fastest". A few seconds later, the Arcada explodes and it's game over.
In the second Space Quest game, you can die earlier than shown above: Walk east on the first screen and fall in to Xenon's atmosphere. Naturally, no one helps you and you are DEAD.
In the third game, you can die on the first screen by cutting yourself on a piece of metal trying to pick it up, or fall into the shredding machine shortly after by not jumping off the Conveyor Belt-O-Doom in time. In the fourth game, you may run into the Cyborg and subsequently suffer Droid of Death on the first or second screen; both also randomly appearing based on hardware speed. Another hardware-based Timed Mission is the formatting sequence at the end of the game, which expires in about two or three seconds on modern computers.
In the second Police Quest game, you can die in a similar fashion as Leisure Suit Larry above by crossing the road in front of the Lytton PD.
At least one of the Kaizo Mario World hacks tries to kill Mario in the opening scene. This naturally started a trend of Platform Hell Mario hacks doing likewise. It's almost mandatory at this point if you're making a Platform Hell hack to try to kill the player in the opening.
In Jak II: Renegade you can fall off the catwalk to your death in the first area within 5 seconds of taking control of Jak.
Volt Kraken/Squid Adler's level in Mega Man X5 starts with an auto-scrolling jet bike segment with a hole so shortly into the level, you can fall into it before the "Ready" on the screen has even disappeared.
In Steamshovel Harry, the tutorial takes so long, by the time you begin the game, it's too late: the world's destroyed just shortly after you can even do anything.
Both Iji and Hero Core have the impossible hidden difficulty "Reallyjoel's dad". The first just overpowers enemies and locks a door, making it impossible to leave the first area. The second locks you in a room with every boss in the game (though with certain late-game upgrades, and free healing if you can kill a boss). The boss room has been completed but only on a tool-assisted run.
It is reportedly possible, if you're seriously unlucky, to have all your dwarfs be thralled or otherwise rendered non-viable for a working fortress before you even unpause after embarking in Dwarf Fortress.
It's also possible to embark on a random hole that goes all the way down, through the world's crust, magma, and into the <spoiler>. You'll see the <spoiler> horde assembled before you unpause for your very first action.
In Super Pitfall The very first ladder in the game? Instant death trap.
AVGN: Nice! Fucking beautiful! The first ladder you see drops you into a pit where you get killed by a bat or a bird whatever it is.
In the Xbox360 version, it's possible to spawn underwater.
In FTL: Faster Than Light, it's possible to suffocate your entire crew by opening all of your ship's airlocks. There's a special message for doing this during the tutorial.
In the Advanced Edition, it's possible (though unlikely) while piloting one of the Stealth Cruisers (ships with cloaking and high dodge rating but no shields) to make your very first jump, and then get your oxygen hacked by an enemy (sucking all the air out of your ship), or a drone fires a lucky shot and destroys your weapons (leaving you defenseless) or engines (which tanks your dodge rating, the only way you can avoid enemy fire due to having no shields).
The Engi B cruiser has only one crewman to start with. The first jump point you visit can result in an event where you lose a crewmember. You don't get immunity from this if you only have that one crewman as such, they will die and the game will end after 1 jump.
It's also possible to die in the Roguelike Ragnarok without taking a single turn, though this requires you to be exceptionally unlucky. Not only would you have to spawn on a stun gas trap, but you'd have to start near enough to a monster so you could be killed before you can regain control of your character.
In Max Payne 3, if you don't immediately haul ass off the soccer field at the beginning of Chapter 3, you will be One Hit Killed by sniper fire. Earlier, the first chapter has enemies fire on you immediately after the mostly-unskippable intro cutscene. If you're on Hardcore or Old School and not paying close attention, kiss your ass goodbye.
Level 7 in Prince of Persia, and Level 10 in the SNES remake. If you aren't holding the Grab key at the start of the level, you'll fall to your doom.
At the start of the Infocom text adventure version of The Hitchhiker's Guide to the Galaxy, you'll die after about your first 20 moves if you don't get out of your house (which isn't exactly straightforward, either).
Border Down has a unique take on the Video Game Lives system. At the beginning of the game, you can choose one of three "Borders": Green, Yellow, Red. Getting hit on Green demotes you to Yellow, getting hit on Yellow demotes you to Red, and getting hit on Red is a Game Over. Additonally, the lower your border, the higher the difficulty. Which means it's entirely possible for someone to choose Red border at the beginning of the game and, either through deliberate action or getting overwhelmed, get hit by one of the first enemies, causing a game over in record time. In the arcade version, this amounts to throwing away money for nothing.
RefleX has a special version of the player ship that is used in the last stage and a half and can be unlocked for use in stage select by completing the game. Said ship is a One-Hit-Point Wonder. This is a game that does not offer multiple lives.
The early Sierra adventure game Wizard and the Princess has you immediately searching for rock to kill a snake. Which one is the wrong one has a scorpion under it that kills you instantly. You have to be on the exact screen which the game doesn't tell you. (The Roberta Williams Anthology manual outright tells you the answer being so brutal) This puzzle was so infamous it was referenced in The Adventures of Jimmy Neutron: Boy Genius.
FEAR's Interval 4 starts with you under fire from all sides on the helipad, which can mean death in two seconds on the Hard and Extreme difficulties. Worse, the Instant Action version has one of the starting mooks spawn with a instant-kill rocket launcher.
In Dark Souls II, taking the wrong path at the very beginning of the game will lead you to a tough mini-boss type of enemy before you have even gone through character creation, meaning you're still a Featureless Protagonist wearing rags with no way to attack but their bare fists.
In the first game, Dark Souls, it's technically possible for that to happen too, albeit a little later during the tutorial, than the sequel. The very first enemy you're introduced to in the first game is a Wake-Up Call Boss, who is next to impossible to defeat without a weapon. Said weapon, you first get it after fleeing the boss and completing the rest of the tutorial.Although... You can defeat the boss prior to the proper tutorial in New Game+ or by having the Fire Bomb as a starting gift in New Game.
The Dungeons & Dragons module Tomb of Horrors places its first inescapable Death Trap at the entrance door, so a lot of parties die by TPK before they even manage to enter the dungeon. You should probably take the hidden real entrance instead.
In Traveller, character generation is done in stages. The first part of each stage is rolling to see if your character is still alive. Yep, your character can die before the game even starts.
This measure, apparently installed to weed out characters too weak to survive the actual game, is optional in the most recent editions of the game, in which it is called "Iron Man Character Generation".
deadEarth seems to follow a mechanic where you have to roll for "Radiation Manipulations", which can actually kill you or cripple your ability to use certain skills before you even start playing. You roll for the manipulations after you've done the rest of the character generation work (i.e., after you've rolled your stats and skills), so you can't create your character in anticipation of how badly he/she will be affected. And depending on how strictly the rules are followed, you might only get three character sheets ever.
Early Magic: The Gathering sets included Game Breaker cards that can let you win the game on your first turn. (There's even a specific combination of cards that, if your opponent went first, lets you win the game before your first turn.) Most tournaments do not allow players to use those cards, and those that do restrict them to one copy per deck.
There are straighter examples that allow you to kill yourself on the first turn or two, such as dumping Phage the Untouchable into your graveyard and reanimating her.