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.
Sounds intuitive but what about rocket propulsion?
Rockets famously take exponential amounts of fuel to reach higher speeds. I'm a layman, but my guess is that this comes from the exhaust speed being fixed. Orbital speed is higher than exhaust speed, so from a frame at rest the rocket leaves behind a bunch of propellant moving in the same direction as it went. That's wasted energy.
Back-of-napkin calculation says that if you managed to perfectly match exhaust speed with current speed, leaving all the expelled propellant stationary, it would only take quadratic amounts of fuels to reach higher speeds. Like the kinetic energy equation predicts.