Starting with FNaF 2, there are minigames that help to tell more of the backstory.
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FNaF 2
You play as Freddy who is following the Puppet in the FNaF 2 location. You can find blood stains on floor, dead corpses of kids and most of the animatronics in the location, minus BB for some unknown reason.
On occasion, Golden Freddy will randomly spawn, but touching him will result in him vanishing. Touching the kid corpses will cause the minigame to end. The game will also have a 1 in 3 chance of ending every 30 seconds, so exploring as much as you can is the only goal of this minigame.
There is a 1 in 100 chance that moving to the left will have Purple Guy ambush the player, which results in the game displaying the message "you can't" and closing the application.
This minigame implies that another set of murders happened, and the tall purple man, who is suspected to be the previous night guard of the location before Jeremy Fitzgerald was hired.
This minigame does not have an official name, but was taken from the words spelling out "SAVE THEM" during the entire minigame.
You play as Freddy inside *Fredbear's Family Diner*. There are six kids who are begging for cake, and a crying child outside of the building. You can't go outside to give any cake to the child, but you can prevent the kids from getting angry by running into them.
Eventually, a purple car will show up, causing Freddy's movements to be slowed down. Soon, a tall purple figure will pop out, which will result in the crying child's tears to grow larger until the kid turns into a gray figure. The tall purple guy drives off, triggering the Puppet's jumpscare.
In this minigame, you can hear the words spell out "HELP HIM". This is implying to help the crying boy outside of the pizzeria. However, FFPS retconned this to have the kid become a girl instead, whom is the daughter of Henry Emily.
* According to MatPat's FNaF 2 theory, which is not debunked by Scott when he said that video got nearly everything right
You play as the Puppet and you see four dead child corpses. You bump into them to give presents, but nothing happens. You then start to put the heads of Freddy, Bonnie, Chica and Foxy on the children. As you add each one, the Puppet's movement is slowed, with the speech also being slowed and made quieter. After that, a fifth child appears, which is obscured as Golden Freddy's jumpscare is triggered.
This game has a point system, where each trip to a child grants you 200 points. When Golden Freddy's jumpscare is triggered, the counter is then randomized. These numbers don't have any meaning other than to showcase Golden Freddy's erratic behavior both inside and outside of the game.
In this minigame, the words "HELP THEM" are spelled out.
This minigame seems to show how the MCI kids got possessed. It's not clear on how the fifth child suddenly appeared like that.
What's interesting is that at the end of FFPS, you see this end screen of the minigame on the monitor. While all the four kids had the animatronic heads on them, the fifth kid in the middle is nowhere to be seen.
It's not know if this is an error by Scott or an intentional lore hint. If the latter is the case, then this may have established another retcon on the fifth missing child. This may not be the case as the dead kids and the Puppet are still seen in the Gravestone Ending.
You play as Foxy who leaves Pirate Cove to greet five kids. You repeat this twice until the third loop where Purple Guy appears, and meeting the kids will show you that all five kids are dead. Foxy's movements are slowed until you get close to the kids, triggering Withered Foxy's jumpscare.
There are no voice to spell out any words in this minigame.
It's not clear where exactly this minigame took place, though given to the purple curtain showing that this is Pirate's Cove, this may have happened at Freddy Fazbear's Pizza, which is where the murders occured.
FNaF 3
These minigames play after finishing any night before Night 6, and which night you clear determines which character you play as. You start out as Freddy, followed by Bonnie, Chica, Foxy and finally a dead kid.
In the first four nights, you play as the animatronics inside the FNaF 1 location. The place seems to be abandoned, full of rats, boarded up walls and rain leaking from the ceiling. You can find Shadow Freddy that displays the message "FOLLOW ME", beckoning the player to obey the message.
If you take your time to explore, you can find hints in one of the corridors that clues you in on triggering the hidden games to get the good ending.
If you follow Shadow Freddy to where he wants you, you'll find that you can't enter the room, causing "ERR" to appear. That's the same "ERR" from FNaF 2. Leaving will result in Purple Guy ambushing the player and dismembering the animatronic. Alternately, if you take too long to get to the room, Purple Guy will attack.
On Night 5 however, Purple Guy will not be able to attack you and you can enter the room where the animatronics couldn't before. There, you will find Purple Guy panicking as you chase him around, and the four spirits blocking the exit. Purple Guy will then run into the Spring Bonnie suit, only for the springlocks to come loose and killing him. The spirits fade away after this.
You play as Balloon Boy with the objective to get all the balloons. After getting them all, an exist will appear, but there is one uncollected balloon as indicated on the counter.
There is a fake wall that by entering, you'll land in a strange place where you can find three dark Balloon Boy figures crying and a tree. Nothing happens if you try to interact with them.
Going to the left will lead you to another room where you find a flashing balloon. Collecting that is important to help the other hidden minigames down the line.
You need to revisit this minigame again to find one of the missing souls after getting the cake in "Mangle's Quest".