The Zelda series typically has the boss of each dungeon's weakness based around whatever piece equipment Link finds in (and sometimes before) it. For example, if you find the Bow expect to be shooting arrows at a weak spot, if you find a hammer then expect to smash some armour and so on. The more recent a game is, the more they do this.
Early on in The Legend of Zelda: Ocarina of Time, Link can learn how to use Din's Fire, a reasonably useful area-effect fire spell, but like the other magic spells, not perfectly necessary. It isn't until he gets to the Shadow Temple that Din's Fire becomes essential, as its the only way to light the torches at once and open the doorway.
We also have Bombchus, which were featured very early on in the game as a minigame in Hyrule Castle Town that you could win prizes from. However, they become a necessity in both the Spirit Temple and Ganon's Castle later on.
In Oracle Of Seasons, the Rod of Seasons can be used to smack enemies out of the way, but it doesn't actually do any damage, which makes it useless...right up until phase 2 of the final boss.
And odd little one shows up in in The Legend of Zelda: Twilight Princess. Near the beginning of the game, if you speak to the mayor of the village, a goat comes flying out of the Ordon ranch, and you either catch it by holding 'A' or get run over. This 'skill' is used in the final fight with Ganondorf, while you're both in animal form.
The goat-catching skill comes into play a lot earlier, when Link is climbing Death Mountain and ends up dealing with charging Gorons in the same manner, officially making this a Chekhov's Boomerang.
In The Legend of Zelda: Spirit Tracks, you can go through the whole game without ever using the Spin Attack. But while protecting Zelda from Maladus, you'll have to do it THREE TIMES with almost PERFECT timing.
In Tales of Eternia, Rid gets the Kyokku skills (Aurora Artes) in three parts. The first two form his two-stage "Hi-Ougi", the high-powered low-HP sort of move most Tales leads get at some point. The last one is completely useless (and unusable) except as a requirement for the final Puzzle Boss.
Inverted in Jade Empire, where many characters comment on a mysterious seemingly unexploitable "flaw" in the protagonist's martial arts, one or two theorising that his teacher included it to trap opponents. Once the nature of the flaw is revealed it turns the plot completely upside-down.
In Trauma Center: New Blood, Valerie drags the operating team to a demonstration held by her friend. Unfortunately, she doesn't realize that her friend became a veterinarian. Any attempts to salvage the situation as having learned something useful are shot down by Markus as being highly unlikely, since human doctors would never have to operate on a dog. Much later in the game, a dog that had been given to them is shot, and they decide to use the skills they learned earlier to operate on it.
Parodied in Final Fantasy IX - early in the game, Vivi, being too naive to understand how to escape a captor, is easily kidnapped from the party. With Garnet standing right next to him, Zidane explains how to successfully resist capture. Later in the game, when Garnet is about to be captured, she only remembers to yell "Let go of me, you scumbag!", to the perplexity of the villains who then proceed to grab her anyway.
The trope is inverted in Disc 3. Kuja's final plan is to use the power of Trance, the game's Limit Break mechanic.
Inverted in Final Fantasy VIII - Ultimecia's trump card is summoning Griever, and finally, Junctioning with it.
Serah is mentioned to have been working as a ground school teacher at the beginning of Final Fantasy XIII-2—and ruling her classes with an iron fist. Much later (literally 400 years later), her Iron Lady skills finally come to use when she chases away a horde of juvenile semi-sentient monsters by yelling at them as she did at her misbehaving students.
Insult Swordfighting, which is required to progress through several points in the game.
A neater example comes at the end of Monkey Island 2: LeChuck's Revenge: at the beginning of the game, you were taught how to make a voodoo doll to defeat Largo LaGrande. At the end of the game, you must recall this skill to defeat LeChuck.
The most amusing example comes from The Secret of Monkey Island, in which Guybrush mentions that one of his qualifications for becoming a pirate is to be able to hold his breath for ten minutes. Later in the game, someone attempts to drown him...and the player has exactly ten real world minutes to get him out of the water. This is also the only way Guybrush can actually die during the game.
In Devil May Cry 4, Nero, not Dante, is the main character of this one. One key feature of his gameplay is his claw, the Devil Bringer. You kill the final boss exclusively with it.
In Mech Assault 2, a key gameplay gimmick was the power armor that could latch onto the body of a mech, hack into it and force the pilot to eject so you could jack it. The last boss is a new, ancient mech that is crawling on it's hands because it's incomplete. You have no mech and the situation looks bleak. The only way to kill it is, yes! Latch onto it and hack it to it reveals a weakpoint.
Shinobi for Ps2: Your sword, Akujiki, will kill you unless you satisfy it's hunger for souls. At the end of the game, the homosexual-looking wizard reveals his true goals all along. Turn everyone in Japan into demons, set you loose against them, and steal the cursed sword once it's full. Oh, wait. You can use the Tate system you've been using the whole time to kill him instantly.
In EarthBound, Paula's 'Pray' ability seems somewhat useless and unnecessary at first. You have to use it to kill the final boss.
Fallout 1 included an optional perk called "Mental Block". This perk would allow you to resist the Master's psychic attacks, with the game describing it as "the ability to tune out any outside mental interference. You must have learned this talent from a passing guru, or from a really late night at a bar."
At the start of Wild AR Ms 4, Jude Maverick is scolded for missing his sword-fighting lessons. Completely averted when he is then given a gun to use for the rest of the game.
In Banjo-Kazooie, one of the many basic moves learned at the beginning of the game is a crouch attack which propels you forwards. However, with the huge amount of other moves at your disposal, especially the other crouching moves, it can be easily forgotten fairly quickly. So, when you're trying to get a MacGuffin by destroying multiple outer layers that can only be broken with a specific move in a specific marked spot, you might just end up scratching your head at what move you need to do for that really low spot...
At roughly the midpoint of the main quest in The Elder Scrolls V: Skyrim, you learn the Thu'um Clear Skies, which you use to reach Paarthurnax by clearing away the freezing mist on the Throat of the World. This Shout gets used again in Sovngarde, to clear the Mysterious Mist Alduin has shrouded the place with and provoke him to battle.
Done in Super Mario World, funnily enough. The one skill you will need to defeat the final boss (punting an item up in the air) is taught to you in the very first level.
Also in Yoshi's Island; to defeat Naval Piranha you need to be good at bouncing eggs off the wall — well, or exploiting a bug early on — so it's lucky that entire boss level drills it into your muscle memory. You know, just a spot of tutorial.
In Mega Man ZX Advent, Model A may be your default form, but you have many better choices for combat. By the time you reach the final boss, you'll have nearly forgotten you have it... much less that it has a homing shot which is practically designed for taking out the boss's multipart shield.
At one point in Golden Sun, Master Hama speculates that Proxian antagonists Saturos and Menardi were able to pass through the Lamakan Desert without Reveal -the spell that the heroes use to pass through the desert, and the Proxians do not possess- because their Fire Clan Psynergy shields them from extreme temperatures. In the sequel, Agatio and Karst, who are also members of the Fire Clan, freeze to death in Mars Lighthouse because the Baleful Polymorph and subsequent battle with the heroes they had been subjected to left them too weak to maintain this ability.