This is my go-to place for traditional, Taiwanese shaved ice. For $60NT ($2 USD), you get a big bowl of shaved ice with 4 toppings of your choosing. The place is quite popular and is pretty much bustling at all hours of the day.
Eastern Ice Store (東區粉圓)
No. 38, Lane 216, Section 4, Zhongxiao East Rd, Taipei