In order: as far as I got on the rewatch) (in order by Wikipedia and may not be in episode order as HBOMAX has it)
Alfred Pennyworth of course new all along.
Robin, Dick (aka Night wing) knew as a child after being taking care of by Bruce Wayne. He was on the search for the killer that murder Dick’s parents and expose himself as Batman.
Bruce Wayne, doctor gave Batman a shot after he was exposed to poison. The blood would surly come back as Bruce Wayne. She came to his machan when Bruce lost his eye sight and said he should take a rest as well as Batman should. (two different episodes)
Out in Crime Alley, Leslie Thompkins (Who helped a young Bruce Wayne cope with the loss of his parents) accompanies Batman to place roses on the spot where his parents were gunned down. Bruce Wayne was well known to do this every year. Leslie would of than known who he was.
The Gray Ghost was told by Batman that he use to watch the show with his dad. He later said the same thing as Batman, in which the grey ghost know knew who Batman was.
Hugo Strange, a psychiatrist. Bruce soon learns that Strange has invented a machine that extracts people's darkest secrets from their minds & transfers them to videotape — & now Strange has proof of Bruce's secret identity as Batman, & plans to auction it to three of Gotham's prominent crime bosses. two-face, Joker and Penguin now learn Bruce Wayne is Batman and dismiss it as well as Jim Gordon also hears that Bruce Wayne is Batman.
a prototype AI known as Holographic Analytical Reciprocating Digital Computer (HARDAC) learns That Bruce Wayne is Batman after Randa Duane had found the batcave. Both were destroy but not after AL created a Bruce Wayne/Batman robot.
While following Count Vertigo's trail, Batman encounters Talia, daughter of the head of the Society of Shadows, who was sent by her father to prevent the capture of a sonic drill that the Count stole. But after his identity is accidentally revealed to Talia, Batman remains off-balance as to where her true loyalties lie.
magician Zatanna finds out Batman is Bruce Wayne when Bruce (as Jhon Smith) had always called her by her name, as Batman did also.
(possible) Thanks to a freak accident during a high-speed chase, the Batmobile is virtually demolished. After Batman takes the car to his personal mechanic, Earl Cooper, the Penguin makes his move & tampers with the Batmobile, putting it under his control. Earl would later learn Batman is Bruce as Batman gave him a new place and said he can by everything from his companies. (Bruce Wayne's companies?)
Catwoman generally discovers that Batman is Bruce Wayne in just about every iteration of their relationship in the comics, animated series, and movies
Barbara was found to be batgirl as Bruce told her he was Batman. In the comics Batman has a relationship with Barbara while she was with Dick which is the reason Dick left to become Night Wing. In Batman Beyond, Barbara speaks to the new Batman (Terry) In which she spoke that her dad knew who she was so it would make since that Jim Gordon knew Bruce Wayne was Batman at the time of Jokers death.
Talia al Ghul removes Batman's mask in her first episode; although she doesn't confirm that she recognized him