Ay-Chung Flour-Rice Noodle, Hsimenting
If you come to Taipei, this place is a must. There are several locations (they've been around since 1975), but I usually come to the one in Hsimenting (open from 11:30 AM to 11:00 PM, I think). These little bowls of noodles are one of the most savory things you'll ever have, with overtones of garlic, oysters, cilantro, basil, and - interestingly - intestines (it sounds nasty, but I bear with it given how good this bowl is). The noodles are thin, and the hot sauce is super spicy (you can add more at your discretion too).
The funny thing is that this place is strictly standing room only - there are no seats or tables. Just grab a bowl and stand around (or squat). A small bowl is only NT$35 (US$1), with a larger one at NT$50 (US$1.50). Yum.
Afterwards, we went around the corner to go grab some dou hua, or a sweet tofu dessert. It was nice and refreshing with a fresh tofu taste yet a mildly sweet and thin syrup to go with it. It's only NT$30 (US$0.90).