I probably watch it 3-4 times a year.
I think a lot of the appeal is the atmosphere (what a club!), the music, the characters (Claude Rains, Sydney Greenstreet, Bogart, Wilson (Sam), all the staff at Ricks, Conrad Veidt as the evil Strasser, and so on - all wonderful little sketches) and the ultra-snappy dialog.
The special effects were first rate as well. The airplane in the final scene as viewed through the fog and hangar door is a wooden cutout drawn in perspective, the mechanics are midgets so as to appear to be in scale.
I consider it to be the greatest film ever made.