I'll explain...
Well, nowadays when I watch anime they're still good and all, but for most I just can't really get into them! People die in them, they end, and that's it! It's like a "Yay it's over. That was a good show" sort of thing. What do I mean by that? Don't exactly mean you were OBSESSED with the show (well in most cases, I guess), but...
-you wouldn't want to miss an exiting episode!
-you stayed interested through pretty much the entire thing!
-you really enjoyed the characters and if somethign happened like one died, that would REALLY upset you (and in some cases, cry)
-if a secret was revealed that you never expected, you would be TOTALLY shocked!
-the ending would come too soon with you going "WHAT?! I WANT MORE! IT CAN'T END NOW!" *throws random object at TV/comp screen* sorta thing...
Here's only a couple I can think of that I liked a show that much:
-Sailor Moon
With the Nephrite thing, I cried when I was little v_v Also being a long series and connecting with the characters, various other scenes in the 5 seasons were pretty recognizable
-Escaflowne TV
Felt like I could REALLY get into this show as well. I was VERY shocked with the secrets being revealed in it, and when certain characters I grew to like eventually, met their fate :/
-Digimon Adventure and 02
Wow... something about these 2 seasons were just... I was TOTALLY into it because the characters were almost as if I knew them, sort of thing. A while back I watched a Bokura no Wargame AMV that recaped major scenes of both season, and it was just so touching! Found another one to the song on the AMV site but... it wasn't NEARLY as good enough! This show was just... something else. Even though it was the dub that I watched it all the way through with! But with it ending, soon made fanfics, fanart, bought figurines and yeah... great!
-Pokemon and Pokemon the 1st Movie
SO into this show when it was out! Sung the theme song (did that to Sailor Moon too earlier too of course^^) and just couldn't wait to see what new adventures Ash would have! Also back in the day when the dub first came out I, I could really feel for that first movie. Characters in a show that I grew to love and the one scene with Mew and Mewtwo fighting... yeah... :/ Of course over time I DID lose interest in the show so it's not like I've seen every ep to-date :/
-Elfen Lied (kinda)
The whole plot of the show and everythign got me wrapped up in it... yeah there was extreme violence and it made it shocking... still didn't need all that fanservice to "lighten the mood though" -_-... but either than that it was one that I was really into because of the psychological storyline and everything.
...think out of all the anime I've seen (and even really enjoyed) these were the only ones I was REALLY into (if you understand my definition of that, of course).