Gamecube is what I have the fondest memories of because it was my console when I got my first job and could buy my own games.
As mentioned already, the startup had a few easter eggs. It's also just a fun and satisfying animation.
What I really liked about it though was how you could access the system menu by holding A on the startup animation. It would transition so smoothly into the system menu which had a cube motif, and the music and sound effects for the system menu kind of gave it an air of...I don't know, a powerful and mysterious force. Plus you might find fun little text snippets about the inserted game or when viewing save blocks on the memory cards.
I have a Wii that I play all my GCN games on now, but...I kind of want to own a Gamecube again just to fool around with the startup gimmicks.