As a tutor, I frequently have conversations like this:

"So we worked out that if when I toss my pen up at 2m/s, it will go 20cm high. How high will it go if I toss it up at 4m/s?"

"40 cm."

"Well, okay, let's check by working through that equation again..."

[We find out that the answer is 80 cm]

"So when I throw it up twice as fast, it goes four times as high, because it takes twice as long to get to the top, but is also going twice as fast."

"Okay."

"Now what if I throw it up three times as fast? How many times as high will it go?"

"Six."

This isn't a misconception about kinetic energy, so much as a lack of comprehension about what scaling is. When students are missing this concept, almost all of physics is more difficult to discuss.