There's a bunch of games out there where one's play piece(s) in the game space represents a person - and any such games are ripe for role-play.
So, Gloomhaven/Frosthaven; Betrayal at the House on the Hill/Betrayal Legacy, Legends of Sleepy Hollow, Arkham Horror, Sentinels of the Multiverse/Freedom Five, Horrified, Aeon's End, King of Tokyo, and even Good Dog, Bad Zombie, all kind of entice you to play the role, because you have a role.
No so easy to do in something like Settlers of Catan, checkers, or bridge, where "you" aren't so much in the game itself.
So, Gloomhaven/Frosthaven; Betrayal at the House on the Hill/Betrayal Legacy, Legends of Sleepy Hollow, Arkham Horror, Sentinels of the Multiverse/Freedom Five, Horrified, Aeon's End, King of Tokyo, and even Good Dog, Bad Zombie, all kind of entice you to play the role, because you have a role.
No so easy to do in something like Settlers of Catan, checkers, or bridge, where "you" aren't so much in the game itself.