Tokyo, commonly referred to as a metropolis, city, or megalopolis is not a city – it's the capital.  Japan is split into prefectures (much like states), and one capital area.  The capital is a large "prefecture-like" region.  For instance, you would not say you lived in "tokyo" even though you lived in Tokyo-to Ichikawa-shi (lit. Tokyo Capital, Ichikawa City).  However for the sake of simplicity the metropolitan area of Tokyo shall be referred to as Tokyo.

Throughout my stay and living in Tokyo I found several itineraries that would maximize the amount to see throughout Tokyo within a limited time – such as a week.  I tested these itineraries with friends and family.  I call it – my Tokyo Death March – due to its strenuous grueling nature to get you to the most sites in least amount of time.