Man, that's such a tough question.
The truth is that you can't single out any one principle as "the" most important. In the past, I've said that animation is kind of like golf. Have you ever tried to golf? Well, I'm not much of a golfer, but I've learned enough to know that there are dozens of important things to remember during a golf swing, and eventually -- after a LOT of practice -- all of these different things become second nature, and they all tie into each other to create one thing -- a good golf swing.
You have to have your toes lined up just right, you have to have your shoulders angled just right, your hips need to be in a specific spot, and you have to swing the club up to a specific height/angle during the back swing. You have to be holding the club in a specific way, and bend your knee in a specific way, and shift your weight from one foot to another mid-swing in a specific way. You have to have one elbow bent just so, and pull with one arm while letting the other drag and steer, and you can't take your eye off the ball, etc.
But at the end of the day, all of those things are really just ONE thing, right? A golf swing.
Animation is EXACTLY the same way. You have to remember anticipation and overlap. Arcs and secondary animation. Force and dynamic posing. Overshoot and line of action. You have to remember squash and stretch and weight & balance and path of action and a million other things.
In the golf swing, if you forget ONE of those things -- say you have your toe pointed in the wrong angle -- then guess what? The ball goes haywire and it's a bad shot.
Animation is the same thing. If you remember everything above, but forget anticipation, then unless the style specifically demands a lack of anticipation, the shot is just plain wrong. It's bad. Because of ONE thing, even though you might have great overlap, beautiful arcs, and the most dynamic poses ever - it's still a bad shot.
We spend a lot of time at Animation Mentor breaking down all the various principles and then showing how they all relate to each other and tie together and morph into one big thing called Body Mechanics, which is basically just the idea of what moves what in the body, and WHY. Overlap, overshoot, force, path of action, etc. -- these are all just different ways of describing tiny facets of what makes our bodies move without falling over, and just as with golf, there's no one facet that is more important than the other.
I wish there was an easier answer, but the truth is that animation requires a serious study of ALL the principles, and the only way to create strong animation is to become extremely familiar with all of them, and -- more importantly -- with how they affect each other.
Best of luck!