TaipeiMetro This is a simple, free map of the Taipei Metro in English and Chinese. It's a little difficult to read if you zoom in because of the resolution, but it works and is accurate.
Taipei Bus Situation Alert This is a bus map - you can search by station and see what the next arrival time of a bus is at that station. It's in Chinese only!
TaipeiPaKing This app is in Chinese only - shows you the nearest parking structures in Taipei and will map a route there for you.
Taiwan Weather This app is a free weather app for Taiwan. Very good for Taipei, though you need to manually enter the date because it doesn't seem to detect it. Simple to use and pretty UI.
Taipei City Guide A free travel guide for Taipei in English including activities and sites, tours, food and drink, nightlife, shopping, and things to know before you travel there. Has transportation and city maps and it all works offline. Recommend.