It started during the development of Street Fighter II: The World Warrior, released in 1991. 35 to 40 people had worked on Street Fighter II over the two years it took to develop.
Noritaka Funamizu, a producer on the development team, invented the combo system that is now so widely known and loved (as well as the subject of at least one internet meme) accidentally.
He says that while performing a bug check during the car bonus stage, he noticed something unusual. He saw that during the punch timing, it was possible to add a second hit, and a third, fourth.
He first thought it was impossible to make useful inside a game, as the timing balance was so hard to catch. The development team decided to just leave it in as a hidden feature.
What they obviously didn’t know was how it would become a base for future titles. It became so popular that it was worked on in the further entries in the franchise. Ultimately, they worked to make the timing more comfortable and made the combo into a real feature- the feature we now see in countless fighting games.