-- A police detective gets kidnapped by the Joker, so Strange and Batman have to go inside the Joker's brain to learn her location. Once they go inside, Hugo immediately fucks off from that mission to find out about the Joker's awful childhood.
-- Separates Scarface, the killer gangster doll, from the Ventriloquist. And then, when the Ventriloquist is living a gentle, crime-free life in Gotham, reintroduces Scarface to him just to see what will happen. Spoiler: Shit happens.
-- Poisons Batman and gets him to believe that he's living in a zombie apocalypse. He then convinces Batman that he needs to bomb the city with an anti-zombie gas. Batman manages to come to just before he destroys the city with a gas that would've created an ACTUAL zombie apocalypse.
-- Takes control of an alien army, and instead of really doing anything with it, just uses their tech to try to download all of the Universe's knowledge into his own head. When the series ends, Strange has gone comatose from an information overload. Batman sees this and is all like "I don't condone this, but I'm fuckin' alright with it."
As a villain in the comics and in the Arkham games, Hugo Strange has always been kind of useless. Sure, he's really smart, but isn't that an attribute most Batman villains have? He wants to know the secret identity of Batman, but who doesn't? It's also hinted that he wants to BE Batman, but if you comb through the histories of most of Batman's rogues, you'll find that a few of them have tried the ol' "Dress up like Batman and see how it goes" gag. But by turning him from a kind of Riddler-lite into someone who manipulates Batman and Batman's villains from a distance, The Batman did something truly special with him.
Oh, and there was that one time that Strange built a robot to fight Batman, but I can't really blame him for that. It was probably written into his contract.