They should restrict certain items on specific champs or make a game mode that does so because having one character be overpowered 15 minutes in is not fun for everyone.

Don't let it happen then.

BOOM problem solved.

How do you stop someone from buying an item?

stop them from farming? and stop dying?

Give us an example of someone that's OP with a single item after 15 minutes and we'll give you a better solution.

shaco boots

I'm going to assume you mean shaco with mobility boots.
Shaco is an early game jungler that excels at making other champions snowball. That means, when it gets to late game, unless he's fed, then he drops off. The easy way to beat him is to focus on not giving a lot of early kills and to ward appropriately. He's not OP, but he's used well in Solo que because people dont' communicate. But buy a few wards on your second back and you counter most of his ganking potential, then have your jungler gank and you win.

garen with any armour pen item (spin = 1.5k dmg, huh?)
garen should not be able to buy infinity!
Adc limited to small hp boosted items not insane tank ones like warmogs

Taric shou;ldn't be able to buy any Armour
supports shouldn't be able to build Tank support (so armour hp) looking at you LULU!

I'm hoping you're being sarcastic Ayerie... because if you are it's pretty funny.

No, it continues to be just sad.