After reading a few answers I still feel like I haven't seen an intuitive answer to the question: why does it take so much more energy to go from 1 to 2 than from 0 to 1?

I have been thinking about it and only been able to come up with something that feels intuitive but not at all precise and I don't know how correct.

When you stand still you may use your surroundings to gain some speed, like by pushing against a wall.

When you have speed it gets harder to gain more speed because the surroundings are (relative to you) moving in the wrong direction, so for every additional unit of speed, it takes more effort to get there.