Im seeing you guys are forgetting another battle where Batman and Superman fought! So they fought more than 3 times it seems!!
It was when Superman was controlled by some Demon and Batman had to go and somehow rescue him! The Demon went in beat the crap out of the other demon that controlled Hell! It was a She-Devil by the way.
The Demon was a regular prisoner of the She-Devil's Hell home. However he somehow escaped and got some help from some peeps of his, found some way to control Superman and took over Hell.
While Batman was trying to figure out to bring back Superman to his senses, He and the She-Devil met! Batman new who she was cause Superman went into Hell before and defeated her plan while he was in there and found what he was looking for and left.
She helped Batman disguised him to go into Hell and fight along with some other Demon who was with the She-Devil's side. They went in their and fought hard but were overmatched.
Later on the book series, the She-Devil gave Batman superpowers to along with a sword and a funny looking skirt (LMAO) Batman fought and defeated Superman enought to free him from the mind control and save the day by defeating the Demon that broke free from the She-Devils grasp
The She-Devil tried to get control over Batman but Superman was there to hit her and throw her back into Hell.
The day was saved and everything was back into normal!!