I can't say I've run up many mountains, but Fusimi Inari Taisha has to be one of the best places to run up a mountain and it makes a nice acronym (FIT). FIT is the head shrine of Inari, the god of rice and is located in Kyoto, Japan.
The main thing about the shrine that will blow your mind are the thousands, and I mean thousands of torii, which are essentially gates without doors that are painted bright orange. The front side of the mountain is lined with steps, which makes for a great workout. Once atop, I found two trails, one going north and one south that help round out the run.
The trails are conveniently marked with numbered signs that help keep you on track. On the sides of the trails are shrines, ranging from a handful of rocks to a mini-temple. The other cool thing are the stone markers that can be found alongside the trail. In this case pictures are worth a thousand words, so check out the pictures below and the run at http://www.strava.com/activities/161041916.
At the bottom, there is a picture of a rabbit, my spirit animal. After the run, I took my kids to a local noodle place that I passed on the way to the shrine. At the end of the meal, the owner of the restaurant presented me two children's books. One of the books concerned animals finding a fur mitten in the snow and turning it into a house. One of the occupants was the rabbit.