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Shitennō-ji is one of the oldest Buddhist temples in Japan, said to be founded (in 593) by priest-prince Shotoku-taishi (who first spread Buddhism in Japan). Only the big stone torii (Shintō shrine gate) is original, the oldest of its kind in the country (1294). It's free to see the gate and to wander most of the expansive grounds. Admission is required to visit the main hall and the five-storey pagoda, which (unusually) you can climb up.
The Honbō-teien strolling garden here (admission extra) is worth a visit.
On the 21st and 22nd of each month there's a very good flea market outside the temple, with antiques and secondhand goods, including old kimonos.