Batman vs L
If you don't watch the anime Deathnote, I suggest you do. L has made me question if Batman truly is the world's greatest detective.
If anyone could out-detective Batman, it's L. L might not be as brilliant fighter like Batman, but i'm asking you who the better detective is. Immagine this isn't a brawl, but a battle of wits. Batman is a vigilante wether we'd like to admit it or not, L is a controversial man of the law that is wrapped in secrecy. Because of this, there's no proof he is or isn't a vigilante and the FBI and even CIA just have to take his word for it.
Immagine Batman is being hunted down for the countless bad things he's done through comic continuity, now immagine Batman has caught wind of L's presence. Now both have to use their deduction skills to find the other first. Both have thier entire lives and careers on the line.
No outside help from the justice league
No Light Yagami to worry about
Just two detectives playing an abstract game of chess.