I like Shrines.
Unfortunately my knowledge of Japan is Kansai and Chubu. I've only been to Tokyo once. I think I prefer more rural Japan than metropolitan areas.
Anyhow, In Tokyo my favourite is a small shrine beside Senso-ji (of Kaminarimon fame)
It is a tanuki shrine.
http://www.flickr.com/photos/hanabidays/312555464
Maybe because of that a favourite with the various businesses in Asakusa. There are many beckoning cat statues in the shrine. Indeed rather than a tanuki the major diety seems to be a beckoning cat.
to digress,
If you ever go farther afield, Jisyu Jinjya in Kyoto behind Kiyomizudera. It can feel very touristy but I was there on a very quite day and had an interesting talk with an old guy who turned out to be the chief priest.
Also in Kyoto is Fushimi Inari Taisya, with it's kilometers of tunnels stretching up the hillside forests. Well worth a visit.
In Ise, there is a shrine by the cave where the sun goddess had to be lured out of hiding. Very small and very remote though. Interesting to be somewhere so outstanding in japan's mythology.
At Japans farthest tip by shinkansen, Kagoshima, in Sengaen I found a shrine dedicated to cats who were brought to Korea in Hideyoshi's invasion to allow their owner to tell the time! (Cats eyes change shape over the course of the day apparently)
Also on Sakurajima, the volcano opposite Kagoshima there is a shrine that is a rotenburo at Furosato Onsen, You have to wear a sort of yukata but you enter the bath through a tori and there is a shrine to Kannon (I think) in the roots of a tree.
http://www.johnharveyphoto.com/Japan3/S ... nPool.html