I like the Kamakura period for a number of reasons. There was a lot of interesting infighting between the imperial court and the bakufu. In the religious world most of Japan's most important Buddhist sects were founded by various interesting characters in reaction to the natural disasters that were a constant during that time. Plus a lot of the best works of midieval literature were composed in that time, especially the military adventures like the Heike monogatari. Makes for interesting studying, though it must have been one of the worst eras in Japanese history to actually have lived through.