The Fushimi Inari Shrine in Kyoto is the best possible example of this, however its always open - so get up early and go before the crowds. I went on one of my first days in Kyoto and after that I went every morning as my exercise. Its a hard haul up to the top and back down and it took me around an hour and forty every day. Some days I paced it out and some I dawdled more but its a good walk and really worth it to go right up to the top.
The Shrine is world famous for its thousands and thousands of Vermilion gates which lead up and behind the main temple buildings. The thousands of shrines in the park are dedicated to Inari the Shinto god of rice. Foxes are important to the story as they are though to be Inari's messengers and you can't go too far without seeing some kind of fox representation.
The gates are donated by individuals and business and can range up to over one million yen for a larger gate. The donators details are inscribed on the gate and as you head up the gates get sparcer toward the top of the mountain. There are multiple shrines and temples dotted around the hike and if you get lucky you can see priests using the temples for offerings and praying.