Have You Played? is an endless stream of game recommendations. One a day, every day of the year, perhaps for all time.
I grew quickly weary of Batman: Arkham City and initially had no interest in Batman: Arkham Knight, but that game being released and then unreleased left me thinking about and eventually longing for the Bat. Batman: Arkham Asylum [official site], the first in the series, is where I choose to return. It's still a remarkable game.
Arkham Asylum's great trick was to take most of the innovations of the modern stealth game and find ways to make them The Most Batman They Could Be. You skulk around in shadow, using vents and high ground to remain unseen, but you're the hunter rather than the hunted. The question isn't whether you'll strike but when and how. Though later games turfed you out into Gotham at large, Arkham Asylum is the perfect setting for that; you're not locked in there with the criminals and thugs, they're locked in there with you - and they know it.
When you string them up from a gargoyle to frighten all their mates, that's very Batman. When you swoop down from a high perch and use your cape to slow your landing, that's very Batman. When you then step seamlessly into to the now famous, rhythmic, gracefully animated punch-punch-dodge of combat, that's The Most Batman.