Ack - I try to keep away from those depressing ones as much as possible, lol
But, one that did make me feel super sick to my stomach was "Kimi Ga Nozomu Eien" (or, "Rumbling Hearts"). That show gave me a headache - just watching the main characters interact with each other was enough to make me vomit.
Another was "Jyu Oh Sei"... kind of a weird choice, but I was not comfortable in the decisions the main character had to make, lol. I was really depressed by the time the show ended =S
Maybe I just have a weak stomach for these kinds of anime, but I seriously do not ever want to watch a tragedy/military anime ever again
One depressing one that I actually enjoyed however, was Serial Experiments Lain. Some parts were sickening, lol (the whole psychological aspect). But the overall story was brilliant xD