What are peoples thoughts on games requiring always online? How does it affect your enjoyment of those games?

I'm currently playing Diablo IV (and having a blast with it) but finding one small gripe which I only think is going to get worse and probably stop me playing it completely in the long run.

My girlfriend is currently pregnant. This means in 6 months time we'll have a newborn. With this in mind I'm expecting to only be able to grab a few minutes at a time to game and even when I think I'll have longer I may end up jumping off at short notice. This means I'll almost certainly come to rely on games which I can pause. Unfortunately this isn't possible with Diablo IV since it requires an always online connection even though I'm essentially playing it as a single player game.

What are other people's thoughts?