For me that would have to be FF8. I always end up feeling close to the characters. Like they're family, or like I was at the orphanage with them. Like I'm the 7th member of the group. The story is just so captivating that the world around me ceases to exist. But I know how to keep myself in the real world. I have my babies (4 dogs, 2 birds, 1 rat) and they always come first.
This is my macaw. His name is Dante's Inferno. He's an evil little bastard, but he's still my buddy.