Personally I loved a good story in a game and without it I have started to get bored when playing games. I enjoy working my way through a level and then having a cutscene as almost a reward for doing so.
I recently played "To the Moon" on the PC. Small game, very little gameplay but an amazing story and it was one of my favourite gaming experiences from this year.
Two of my favourite games Persona 4 and the Mass Effect trilogy have great stories and use them really well to have you connect with those characters and universes in a way that I never have before with other games.
As I said in the other thread, Nintendo games have started to feel a bit hollow for me. Following the same old things time and time again. They were fun when I was younger but I think progression is needed.