i think it will end with him becoming THE BATMAN,not just "Bat-Man"
let me explain...
in the first film,obvoiusly he's just starting off....he's a bit of an urban legend,some people know about him,some don't,but (as the kid in the Narrows tells him) everyone is talking about him..he's got the "buzz" of being something new....he's also not 100% on top of his game yet - he gets beat by Crane at one point,and causes massive property destruction via Tumbler and train (with help from Gordon)....
in TDK,he's more skilled,and the criminal underworld fears him,but more as a man who can beat the s**t out of them than anything else...they all more or less consider him a vigialnte in a suit...Chechen brings dogs to deal with him....Maroni feels confident enough that his goons can beat Batman that he stays in his seat in the club while Bats goes through them all...Maroni tells him that they know he has one rule,the Joker doesn't,and nobody's gonna tell him nuthin - the mob is more afraid of the Joker than Batman...he's also driving around indoor galleria's and such on the Batpod,in full lighting,letting people get a look at him,so the ordinary citizen can think of him as a guy in a suit too...reese also goes on the air and threatens to reveal his identity,once again letting the average Gothamite know this is a man,with a life and identity under there...and,of course,the Batman imposters taking on crime themselves,figuring they can do just as good a job as him since they know he's a man in a suit ("What gives you the right? What makes you different from me?") ....so in essence he still thought of a just a man in a suit - a tough,nealy unbeatable man,but just a man....
in Batman 3,i think he'll finally become the "legend" Ra's said he could become...something will happen that will turn the criminal's into the cowardly,superstitious lot...ironically,Batman's embracing of the idea that he killed people to save Dent's reputation may ultimately work in his favor towards this goal...like Maroni said,they all knew of his one rule of "No Killing" - if they believe that's shattered,well then that would put fear into them as they believe he WILL kill them if necessary, a huge difference in the psyche than just knowing you'll get punched in the face and sit in jail til you make bail...so the criminal element will be afraid of him again...
as for the rest of the world,i figure it will be something involving both the threat of the film,and perhaps a new,more evil/animalistic looking suit....at the end he should be the type of character that people talk about in hushed whispers...that no one wants to get a picture of,or hope they see...he should be the type of figure that they PRAY they never see!
something that lurks in the shadows of the buildings,the grime of it's alley's....something the people of Gotham know destory's crime and criminals,but also frightens them into staying away too...is he human? is he an animal? is he a monster?
he's the Batman....a Legend....and that's what he should be at the end of Batman 3,and it should end with him taking his place as the Legend of Gotham City...