'Mad' is not the correct term

Discovery

Izuku's quirk did not grow over night, sadly. Instead, it developed over months. How do we know this you ask? Because of his speech. 'His speech?' (That's my impression of you by the way!) Yes, his speech. It used to be straight forward and simple (May I have one of those, please?), what you'd expect from a normal four year old, but as the days and weeks flew by, it began to become a bit... jumbled.

At first it was just skipping over a few words, not really noticeable, but then he began mixing and mashing the words meanings. He began to speak backwards, sometimes without noticing, sometimes on purpose. And when some foreign sounds came from his maw, he claimed to have been speaking upside down.

Now, if you were a single mother (or father, one must be aware of the fathers), you'd be a little worried, your kid who seemed to be fine is now getting basics reversed. Or maybe a lot worried. Depends on the person, and if your the same person as Midoriya Inko, you'd be a lot worried. So what's your first course of action? Take him to the hospital. But, alas, hospitals can only do so much, so you go to your second course of action. Take him to a quirk specialist. I assure you, you would be thinking, 'He is around that age, maybe it's nothing, just a drawback that can be managed.' Right? Well, you'd be perfectly logical if you did.

At the quirk specialist (after finally booking an appointment) Midoriya Izuku was undiagnosed. The claim the doctors had to this undiagnosss was the fact that it was to early to tell. 'Come back after a couple of months or if any major quirk like symptoms show.' They had said. 'Useless.'
You stole the words right out of my mouth!
As they returned home after another quest to answer a mother's worry about her son, she gave him a quick fever run down. 'She's checking his temperature.' Right! Correct! You're following along splendidly! Her son had no fever,  nor did he say he was light headed. He could breathe properly and was not flushed, so she whipped up some dinner and sent her son to bed.

Over the next three months (it was now August), his speech did not become any worse, and he began talking as society expected him to(only under intense concentration).
His speech did not become any worse, but his hair did. It used to be tame able through hard work, now it was absolutely feral. No brush, hair solution, or exorcism could get his hair under control. Finally giving up after breaking the fifteenth brush, Inko decided her son would simply have to wear a hat. While it wasn't the solution she preferred, she was happy that the hair remained soft. It would be a curse to have scratchy hair pushed down on your neck all day. (Trust me, I know.) That weekend she and her son spent the entire day shopping for a suitable hat. The problem was that Izuku was always drawn to the bright colorful hats that did not mix well with his hair. They soon came to a compromise. He could have a tall Top Hat if his mother could choose the color, and obviously she choose black. The color that goes with almost everything. Walking out of the store with a Five Year Old in a top hat was once experience Inko thought she'd never have to go through.

At school, the children loved Izuku's hat at first. Then came the rumor that he was quirkless, a few denied it could be possible, 'Look at his hair! His speech!', and stood by him. Others... well, you know how the story goes. Only this time, there's a slight difference. The bully? His name was Yato Ayato. (Not very creative, I know, but I'm not good with names.) His quirk was called Luck. If you have him a five yen piece he could make any insignificant wish come true. The more money, the larger the wish. He could never use it on himself though. His second in command was Haruchi Kondo. A large guy who only followed Yato around because they were childhood friends. His quirk was Reshape. Any object he touched he had the ability to reshape, however, it could only be made out of the same amount of material as the original form. (Worded poorly, it means if he touches something like a glass cup he could change it into a plastic cup, but it could range between the same size, shorter but wider, or taller but skinnier. A book can be changed into a scarf, the length and thickness deepening on how thick the book was. You get it? 10=10. 10 can be changed to 5+5 or 3+7, but it still equals 10.)
Those two basically ran the Mido-Hurt Squad. Fortunately, there is always an opposite.
The Mido-Protection Squad consisted of,

Shinso Hitoshi. Quirk: Brainwash; When Asked a question by the user whoever responds in under said users control, but the user can only command direct actions.

The third in command was Shoji Mezo. Quirk: Dupliarms; User has multiple arms and can change the attachments at the end of each arm into a body part. Each tentacle attachment takes the same damage as a regular body part would.

The second in command was Bakugk Katsuki. Quirk:Explosion; The user secretes a nitroglycerin substance and uses the heat from their palm to create explosions. Can cause wrist damage.

'Why would they protect Izuku?' The answer to that is simple. He is a very strange child. Why else would these three be drawn to Izuku? He was just like them after all. Different. That's also why they were the first ones to notice when he was becoming a bit pale. They told Inko of course, and she took him to the hospital again. 'Nothing was wrong with him' They said. Which meant she had to go to the quirk specialist.

Which meant tests. They took blood samples, x-rays, and even looked up past quirks that resulted in pale faces. The real give away to the answer, was in fact, not one of the tests. The answer was given to them when the Doctor took Izuku's hat off and placed it in a box, only to turn around and discover it was I tip of Izuku's head. 'How did it get there? Is that his quirk?' The answer to the second question, is a maybe. The answer to the first question, is an 'It got there.' Having his suspicions, the Doctor removes the hat again, and put it back in the box. He turned around for three seconds exactly, then turned back around. The hat was on Izuku's head.

'Well I'll be.' The Doctor thought. 'I must write this down.' Naturally, he reached towards his breast pocket, only to discover his own was gone. He looked around, but it was no where in sight. A slight tap on his side told him to turn around, and there was Izuku, hat in hand. He put up one finger, and reached down inside the hat. A couple seconds later and out came a brand new ball point pen. Having new theories, (as people often do if they're smart) and new tests, it was discovered that Izuku had some sort of attachment to the hat, and when ever he needed something, he need only to reach inside the object in his head. The doctor named his quirk 'Anytime' and sent them on their way. Inko still wanted to know why her sons face was turning pale-ish white, but just assumed it was a draw back.

The bullying stopped, though not by much, and Shinso, Shoji, and Bakugo continued to be Izuku's friend. Even if he was a bit odd, they soon could understand his way of speaking. Just a little bit.

It wasn't until Izuku's sixth birthday that another strange anomaly happened.