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In the third world, it is possible to land the airship on the entrance to Guido's cave. The airship is then lost, since walking onto that square will cause you to enter the cave. The airship is, of course, necessary to beat the game. It is also possible to lose the Hiryuu this way, and lose the black Chocobo by landing in the center of the Mua forest, but losing those won't make the game unwinnable.
Final Fantasy VI
In the World of Balance, there's a tiny patch of land to the right of Nikeah just big enough to land the airship on. But if you land there and go into the town, then there's no way to get back—the town blocks you from reaching that patch on the World Map, and all of the exit points within the town spawn you to the left side. There's nothing preventing the player from saving the game in this situation. And without the airship, the game is unwinnable. This example is especially sinister when you consider that the primary reason a player is likely to take the airship to Nikeah is to pick up Mog's Water Rondo dance, which is Lost Forever after a brief window of game events.
The original Japanese version had a bug in the World of Ruin that would send you back to the World of Balance. You can't come back to the World of Ruin. This was fixed in all subsequent releases.
Final Fantasy VII
If the player saves just as the battle with the Diamond Weapon would normally begin, then reloads the game, the Diamond Weapon will have vanished, and the game will be unwinnable, due to the cutscene where the barrier over the Northern Cave is destroyed not occurring.
Final Fantasy VIII
There's a point where the characters have to stop a missile launch, but to get the codes you have to help out some enemy soldiers. If you refuse to help them, then you cannot change the missile launch, and it's possible to save after refusing to help, which means you'd have to start the whole game over if you don't have a back-up file.
After having access to chocobo, it is possible to ride a chocobo through shallow water onto a barren island in world map, get off, and get stuck. Consider that the world map is the place where saving anywhere is possible, it is entirely possible for a player to save here and essentially get stuck for eternity.
You better have enough healing items on you when you reach Ultamecia's castle, because once you enter, you can't get out. To make matters worse, you don't have access to your abilities until you beat a certain boss.
Final Fantasy X
In Final Fantasy X, there is an airship battle against "Overdrive Sin" near the end of the story. Meaning that, basically, you can only inflict damage with magic because of how far away the target is at the start of the fight (the only one who can hit it with melee the entire time is Wakka). Sin's Overdrive gauge will fill up a bit every turn, and once it's full, it'll use an attack that causes an automatic game over. Doesn't even inflict damage, because it attacks the airship itself, (and razes half of Spira) so even if you've powered your characters up enough to survive both the attack and thefall, you just can't reach Sin. If you have only one file, you save right before the battle, and you're underleveled or have been neglecting to train Wakka or your magic users (primarily Lulu at that point), then you're completely screwed.
There is one way and one way only to get out of this situation without having to restart the entire game from scratch: Yojimbo. If you had the foresight to bring him, his Zanmato Overdrive is capable of a one hit kill against absolutely every single enemy in the game regardless of how low your personal stats are. But his willingness to use Zanmato is directly influenced by how much he likes you and how much money you have - if he hates your guts and/or haven't got a penny to your name you really are finished.
The siege of Bevelle immediately after fighting Evrae is also like this for some people if they're under-leveled, although having some equipment abilities can make it drastically easier (Stonestrike, Fireproof/Fire Eater).
Final Fantasy Adventure
Final Fantasy Adventure requires the player to know in advance that they need Mattocks and Keys. The need of Mattocks is eventually eliminated by the Morning star weapon; but if you don't have enough keys before entering Dime Tower, or if you save when you don't have enough keys and can't go back...then you can't beat the game.
But that's not all. It is entirely possible to miss the Morning Star. You receive it as a reward from a boss battle; but the battle is optional, and getting to it requires traveling through a frustrating maze.
It's possible to save in a room that you need a key to get out of, where no key-dropping enemies spawn.
Note: Resetting the game causes all doors to close. If you have a key shortage, then even going back the way you came might be impossible.
It is possible to save at any time in this game through the menu instead of using save points. In at least one room of the airship around the middle of the game, it is possible to be permanently stuck if you save in that room then load that save, you will appear in a fenced off section of the room that's normally inaccessable with absolutely no way of getting out.
Final Fantasy Tactics
Final Fantasy Tactics has one where in Riovannes Castle before the final battle with Wiegraf. The first part of the fight is 1v1 with Ramza, and if you don't have the right equipment and/or skills, it's basically impossible to win. If you save right before the fight, you have no chance to go back to level grind or purchase equipment, so if you don't have those skills and equipment, you're forced to restart the game.
Owing to the games mechanics, you can still be screwed even if you have 20 levels on him.
When Wiegraf is beaten here, he turns into Velius, and Ramza's party catches up to make it a relatively standard boss fight. However, Velius is easily the most difficult enemy in the game. It's entirely possible for Ramza to be easily capable of beating Wiegraf 1v1, but for Ramza plus the rest of the party to be woefully incapable of beating Velius. The end result is the same, unless Ramza can survive long enough against Wiegraf to buff himself into a One-Man Army.
And finally, there's the Riovannes Castle Roof battle after the Velius fight. You have to keep Rapha alive and bring at least one of the three enemies down to end the fight. Sounds easy, right? Well, two of said enemies can use skills that can always successfully land a One-Hit Kill or a Stop ailment. The other one is very strong with area-effect attacks. All of them also has high Speed so they will take their turns first. Your units barring Ramza are all placed in the far side of the roof. Rapha also likes to be brave, making her exposed to the enemies. So unless you have foresight on what is going to happen (by having Thieves, Monks, or speed and move boosting accessories and skills), you're definitely screwed.
In the battle against Gafgarion at Lionel Castle, Ramza is trapped on one side of a wall with the primary villain of the piece, while the rest of his team is on the other side with an assortment of mooks. The way you're supposed to meet up with the rest of your party is to open the gate in the middle of the wall using a switch; however, the ground is laid out so that it's possible to get Ramza to the top of the wall at one corner. With a certain movement skill, Ramza can jump down to the main battle area and help out his teammates without unlocking the gate (given that it's possible that the mooks will outstrip your party members on their own, and that Gafgarion will do his best to keep Ramza away from the switch, this is a reasonable thing to try). However, although that movement skill will allow you to jump down, it won't allow you to jump back up, trapping Ramza on the far side of the gate; if Gafgarion hasn't died, this renders the battle unwinnable, as he can hide in one of the far corners, just out of range of the spells available at that point.
Dirge of Cerberus
It is possible to get autosaved at the start of boss fights without enough ammo to win or items to survive, most notably against Weiss, because he's too fast and strong to fight hand-to-hand. Some other boss fights put a vending machine in the boss arena, as little sense as that makes. The battle against Rosso the Crimson is one such battle.