Harry could disassemble the world and the stars into computronium - in fact star-lifting crossed his mind when he heard the first part of the prophecy. EY stated that there would be no AI analogy, and magical intelligence amplification seems more plausible anyway.

A different route to a mini-foom is that one can make luck potions. Then gamble, get money, recruit people who know how to make potions. Now you have huge amounts of luck potion, and provided your thought process is fairly random, you will always find the right answer (e.g. opening a random book at a random page happens to provide exactly the right insight). Routes to magical IA:

Luck potions.

A thinking hat - like the sorting hat, only it uses your brainpower to help you solve problems.

Efficient use of memory charms to spread insights rapidly through a group of researchers.

Use of telepathy to create a group mind.

Potion of thinking, made of e.g. ground-up crossword puzzles.