There's certainly story happening just don't expect much.
As far as I've read in the RAW
Since Ash's job is dealing with the hero a lot of his attention is on that but things happening in the background sometimes get brought to his attention.
Like the demon lord vs goddess fight mentioned here goes awry because of his survival in the upcoming tutorial section.
Although one thing I can foresee certain people to start making a fuss about
is the amount of little girls he get in his harem
as of latest raw I've seen his harem include (in order of joining)
Luna the dark elf
Eris the cat maid assassin (who got her revenge on the rogue monster that destroyed her village later)
Caroline the succubus
Cocolulu the slime inn girl
Ricola the succubus (Caroline's little sister, who's a lot more accepting of becoming a demon. Call Ash 'onii-chan')
