Anything with access to the Dark Mage/Sorcerer class or Shadowgift has access to one particular tome: Nosferatu! Nosferatu makes the entire game a cakewalk on three difficulties, and is almost required on Lunatic+ due to blatant cheating on the computer's part. And as stated below: the Ruin tome, coupled with skills that boost Critical Hit (Wrath/Zeal/Focus) and lowers Critical Avoid (Anathema), is certain death to almost anyone in your way.
An even better tome is the "Aversa's Night" tome, which has the same effects as Nosferatu but with 15 attack instead of 9. With forging, you can increase this up to 20, which is more powerful than a vast majority of legendary weapons in the entire game!. It has less durability than Nosferatu, yes, but with the Armsthrift skill mentioned below and Hammerne staves being purchasable with Streetpass abuse as mentioned below as well (and by randomly from the spawning merchants found in endgame locations), this becomes a mere technicality.
You can also combine this combo with the Sol Skill, which causes 50% of the damage dealt to heal you as well, this stacks with the Life Drain effects from Nosferatu/Aversa's Night for 100% healing from dealt damage.
Now add some supportive skills like Pavise (Skill% chance of halving Axe/Sword/Lance damage taken), Vantage (Always attack first whenever your unit has 50% or less health), and Wrath (+20% crit rate whenever your unit has 50% or less health), and you've got a monster in your army.
The Avatar is this to a ridiculous degree, with his/her easy access to almost every class in the game allowing him/her to collect a ludicrously powerful selection of skills.
And he/she can very easily pull off the above combo of Sol+Pavise+Vantage+Armsthrift+(Wrath/Anathema/Astra/Vengeance/LimitBreaker) with enough grinding.
If the Avatar is capable of throwing off most attacks with high evasion, then you can probably trade off some of the other skills for more damage increasing ones. When Ignis is paired with Limit Breaker, it can potentially add an average of 20+ extra damage when Ignis activates. That's like at least an additional 25% of the maximum HP on most enemies (where 80HP is the max, not including bosses at higher difficulties).
Paragon. Obtained in Lost Bloodlines 3 DLC, this skill is like Veteran, except for two main differences: double the experience gained and used even without being Paired Up. Needless to say, it's very easy for everyone to become a Disk One Nuke. Moreover, the skill can also stack with Veteran itself.
Avatar's child Morgan can easily become one as well, especially if you pair Avatar with a second gen character who already received boosts from their parents. Or really, Avatar children in general thanks to their access to every generic class in the game. While having a Third Gen Morgan is pretty amazing, you can opt to have two Morgans on the team, by having your Avatar marry a character that can sire a child.
A Sumia!Lucina can be terrifying from the word go, even with very little grinding, due to her ludicrously high Skill and Speed Growth.
Aggressor, a skill gained from the Dread Fighter DLC class. It grants a flat +10 to attack power to every attack of the character with it on the players phase. This is a huge amount in the main game, and considerable even in the hardest DLC.
The Galeforce skill, learnt by Dark Fliers. It gives the user another full action after defeating an enemy once per turn. Remember Canto? This is basically a superior version of that skill, which is saying something considering howgame-breaking the ability to re-move after an attack really is, let alone getting a guaranteed 2 turns at once. The applications of this skill is uncountable, such as killing 2 threatening enemies at once, accelerating your completion of map objectives, better positioning, performing a hit and run strategy... the list goes on. And thanks to the way the game is set up, offense is basically not an issue at all. Throw in buyable Rescue Staves and Dance support to further increase their versatility, and you get a gamebreaking combo. To tip the scales in your favor even more, you can get the first generation characters to pass the skill on to their kids, being the only way any male units can obtain Galeforce.
Buyable Rescue Staves. Rescue is an item that many beginners would consider no more than a grinding item, thanks to its huge exp gains for staff users and usability at E Rank (which is a huge factor of its usefulness). And that is only a simple application of its full capability. Fire Emblem, being a turn based strategy game reliant on having proper positioning, is made much easier by Rescue, an item that allows you to manipulate your positioning. In past games, Rescue was a very rare item that reserved for special turn cutting and game changing decisions. However, now that the staff is spammable, you can use it to send in your units to do any kind of risky strategy and simply use Rescue to clean up the risk, earning massive exp in the process. Consider that you have at least 2 competent staff bots right from the start and the staff is buyable after Chapter 13 (you get two of them before then), and you get the drill. It says something when Apotheosis is less "challenging map that requires strategy" and more "kill something, use Rescue Staff, repeat."
It also takes much of the pain out of escort or recruitment missions by letting you corral the VIPs into a defensible position, surrounded by your strongest units, rather than splitting your army to protect them individually.
Considering that everyone who has access to Dark Flier also has access to Falcon Knight, it gets very easy to combine the aforementioned Galeforce with Rescue Staffs.
Certain SpotPass units, such as Sigurd or Alvis are easy to obtain (oh, look 17000 in-game gold) L20 promoted units with a bunch of skills and very high stats, not unlike the Black Knight in Radiant Dawn... except they can be reclassed to lower levels, and don't leave your party.
SpotPass teams in and of themselves. More accurately, the ability to buy items from them. These include otherwise pretty rare and hard to replace things that only appear regularly in lategame, such as Silver, Brave and Killer weapons. "Eirika's Blade" and "Celica's Gale" are also notable purchases as they are fairly inexpensive items, both have a brave effect, and anyone with either C-Rank sword or C-Rank tomes weapon skill can equip them respectfully (And the Avatar's default class, Tactician, can very quickly train himself to wield both). And Spotpass teams can be summoned infinitely, so if one is patient enough, they can pretty much buy a large stock of these items at their own leisure and not need much else for the majority of the game.
And if a Female Avatar marries him..... Morgan could easily kill gods at that point.
Two words: StreetpassAbuse. Want to buy more stat increasing items without having to wait on a merchant, Thief or Barbarian related Risen to show up? Want to purchase those mysterious barracks items without testing your luck on event tiles and barracks? Can't buy Master or Second seals yet? All one would have to do is equip those items on their streetpass team and save that team and file on two or more 3ds' (which requires a game cartridge to work), rinse and repeat. Time for some truly mind-boggling item farming to the eleventh degree, and it even works if that file is deleted immediately afterwards.
Second Seals, which not only change the class of one of your characters, but also have them keep the same class by resetting their level back to 1, allowing them to keep all of the stats and skills they had acquired up to that point. This means it's possible to max the stats of ALL of your characters if you are willing to do enough grinding. Reeking Boxes also count, since they're rather easy to get and it allows you to pretty much farm Risen for experience and gold (on Normal difficulty at least, you won't be able to profit on higher difficulty levels). Adding to this, for the first time in the series, is the option to skip the entire enemy turn by mashing start, making grinding infinitely quicker. Just place a character in the middle of the map and skip. Allows for grinding Supports, too.
A certain combination of rather monstrously powerful skills and spells. Vantage allows you to always attack first if your health is below half. Wrath increases your critical chance if your health is below half. Vengeance increases base damage more the lower your health is. Combine all of these with a forged Ruin tome and it's perfectly possible to one shot the final boss on Lunatic difficulty.
Wanna boost the above? Use the very simple Pair Up! Not only does it give stat boosts (which can get bigger if the two characters Support), but it also gives the chance for the second character to also attack, as well as negate a attack his/her partner might take. An S-ranked couple can potentially solo maps even on Lunatic difficulty. Losing one character is not even an issue when you get an obscenely overpowered character in return.
The Mercenary starting skill Armsthrift, which allows you to use a weapon withoutexpending one of its uses when it activates. More importantly, it has a rather high activation rate: twice the Luck stat of the unit who's equipping Armsthrift. With Pair Up and/or sufficient grinding, it's possible to raise your Luck to 50—which means your weapons will never break, one of the units who can do this easily are Donnel!Owain or Morgan with the right stat buffs. All those legendary or expensive forged weapons and tomes which were Too Awesome to Use? Feel free to combine them with any of the above Game Breakers for even more unstoppable power. However, it's worth to note that Armsthrift only becomes genuinely broken when your Luck stat gets high enough. Before that, it's merely useful, but not game breaking by any means. This makes it awesome for postgame, but balanced during the main story.
The real gamebreaking application of this skill comes from pairing it with a weapon called Leif's Blade. It simulates the skill Despoil, which gives you a percentage chance of getting a Small Bullion (worth 1,000 gold) equal to your luck whenever you kill an enemy on your turn. At 50 luck, you will activate Armsthrift with every attack and the sword's skill nearly every other kill, on average, allowing for infinite gold even on higher difficulties where Reeking Boxes cost more than you will make off of clearing a map with one. It can be grinded on low-level Spotpass teams by a single character who can solo the map with ease. Alternately, an Inigo with Chrom as his father, Spot Pass Ephraim, Spot Pass King Marth, and DLC Celica will only need 45 Luck and the "Rightful King" skill to abuse this trick.
Gregor's class set is pretty good for a first-gen unit. He starts with the aforementioned Armsthrift, and can pick up Vantage, Swordfaire, Axefaire, and Wrath through careful reclassing. With a unique axe, he can even use the Vantage-Vengeance-Wrath combo that normally requires the Avatar or a second-gen unit to pull off. And did we mention that he starts with Armsthrift, which lets him use that unique axe as much as he wants with enough Luck?
The skill Limit Breaker. It extends all skill caps by 10, and this makes wielding legendary weapons a breeze since all promoted characters that can get Armsthrift will have a luck stat of at least 50 with it equipped, and all SpotPass and DLC units can learn it. Other than that, it makes the character in question much more powerful and increases the likelihood of their skills activating.
Chrom and Lucina have a skill called Dual Strike+ which essentially increases the Dual Strike activation rate by 10. While it's not flashy, with the right spouses and skills, a 100% Dual Strike chance becomes possible which combined with other gamebreakers like pairing them up with someone possessing a Brave weapon can potentially allow the pair to attack up to 12 times. Combine with Vantage + Vengeance on the lead unit (guarantees potentially 6 hits, with two potentially being boosted by Vengeance, before the enemy even gets to act) and watch the destruction unfold.
Most of the children are this when their mothers are paired with the right spouse. A few examples:
Pairing up Sully with Vaike will give her daughter Kjelle insane growth in most stats, but especially strength, health and defence, and if you keep her a Knight->General then she will be one of the most staggering tanks the franchise has EVER seen.
If you pair Nowi with Kellam, then her daughter Nah will be an unkillable dragon-shaped fortress that hits like a truck from near and far thanks to being a manakete.
Want Lucina to be really scary? Pair Chrom with either Sumia as mentioned above, to give her the Dark Flier path for Galeforce, OR pair Chrom with Olivia to give her the Myrmidon line so she can get Astra from Swordmaster and/or Lethality from Assassin.
Considering how strong his mother Cherche is, its no surprise that Gerome can be absurd. Pair Cherche with either Vaike, or if Vaike is with Sully, pair her with Frederick. Gerome will then be a mobile battering ram that can 1-shot almost anything.
Severa can be ridiculous. Pair up her mother Cordelia with Stahl. Severa now has sky high growth in health, strength, skill, defense, speed AND resistance, making her one of the game's most broken lighting bruisers.
There are a LOT of ways to get Avatar's kid Morgan to be one of these, but we'll list what might be the biggest. See the above formula for making Nah a huge tank? Make an Attack oriented male Avatar and pair him with said Nah. Morgan is now an unstoppable dragon who WILL obliterate everything that dares to fight her and she will probably never take damage from ANYTHING.