Boardgames: I love playing social deduction games like Mafia and The Resistance with a big group, though they aren’t everyones cup of tea (perhaps I enjoy lying too much?)
Video games: We play a lot of Quake Arena in our office at work with a big group of people, also Genital Jousting with a smaller group of closer friends is silly fun.
Any of the Lego games are pretty awesome for local co-op.
I’d also recommend Castle Crashers, it’s a hack and slash / beat 'em up style game that supports up to 4 players locally.