Easily, hurricanes, tropical storms or the remnants of tropical storms produce the most rainfall across wide regions. Rainfall amounts can top out well over a foot over the course of a day or two in areas near where the storm makes landfall. In our area, the remnants of hurricanes and tropical storms have produced over 6" of rain on many occasions.
But if you're talking about a storm that produces the greatest volume of rain in a short period of time, it's most commonly the thunderstorm. While not all thunderstorms produce rain at the same rate or volume, some can put down rain at the rate of two inches an hour, or more (although that rate of rainfall may only last for closer to 30 minutes).
Keep in mind that heavy rain bands within tropical storms and hurricanes are also capable of this sort of heavier rainfall rate at times, but thunderstorms are more common, especially in our area, which is why I'm giving them the advantage in this category.
