I just realized that I didn't actually suggest the anime that immediately came to mind:
Full Moon wo Sagashite.
It's one of my all time favorites. The first season does have a slow pace, and may not get any tears from you, just laughs. The second season though really picks up the plot and emotions surge. The ending is just spectacular. If you make it through every episode without shedding a tear, you're probably a sociopath. As you can see from the link, it has a very high rating. That's because it's awesome.