I used to think something could rotate around more than one axis at once. Imagine a pipe sitting in space with some jets on it. Two opposing jets on the middle angled tangent to the curve firing equally would set it rotating around the long axis. Two opposing jets on the end angled perpendicularly to the pipe would set it rotating around the short axis. I thought you could do one of these and then the other and get something that was rotating around two axes at once. Then in high school I was writing some kind of space program that had objects and I needed a way to represent their rotations. Each object was fully rigid and had a position (x, y, z), a velocity (dx, dy, dz), and an orientation (ox, oy, oz), but how should I represent rotational velocity? Each one would be a vector (rx, ry, rz), but what order would I apply them in? Did that matter? How did rotational velocities around multiple axes interact? At this point I went to talk to my physics teacher, who explained that there's no list of these velocities and when something would add a new rotation to an object it instead combines with the existing rotational velocity. Which is why gyroscopes work.

I'm curious whether I could have gotten the physics to work out if all rotation was independent, and what else would be different about that world.