But my all-time favorite show has to be Monster, from Naoki Urasawa. At the very least, I can say that I don't like a lot of anime (I'm generally biased against them), but this being at the top of my list speaks volumes of it's quality. It avoids all of the issues I normally have with anime (such as trying to juggle between funny and serious, super-powered teenagers, a variety of filler episodes (such as beach episodes), far reaching plot points and a general sense of things that don't quite fit together). It is a cohesive, filler-free, incredibly realistic man-drama about a Japanese neurosurgeon practicing in Germany after the fall of the Berlin Wall, who accidentally saves the life of a ruthless murderer.
Unfortunately, it was removed from youtube, but the above link has all of the episodes of the series.