Actually, in regards to producing tremendous amounts of rain, Category 1s and tropical storms pose the greatest danger.
Here is an excerpt from
Encyclopedia of Hurricanes, Typhoons, and Cyclones by David Longshore.
Tropical storms generally contain enormous quantities of precipitation. .......examples are given.........
Meteorologists contribute this destructive phenomenon to the tropical storm's evolving nature. In a mature hurricane, moisture levels are rapidly decreased by the heated process of evaporation, thereby reducing the amount of rain the storm actually carries with it. But in a tropical storm, where the convective cells are still organizing, the moisture-laden air is not rising fast enough to complete the evaporative cycle. Vast amounts of rain begin to collect in the edges of the cells, darkening the storm's pale white color. This precipitation will continue to gather until the tropical storm either reaches hurricane force or makes landfall.