One of the greatest, and worst, things about Taipei is its basin-like structure ringed by mountains. On hot summer days the heat and smog linger for what feels like an eternity and even strong winds can't blow it all way. Yet go up any of those mountains at night and you'll be rewarded with a spectacular view.
Enter one of the all time classic things to do in Taiwan: The night view. Despite some rather intense discussion about if "night view" actually means anything in proper English, it's definitely what it's called here.
Elephant Mountain at night in 2013.
The most popular spot to take in the city at night is Elephant Mountain. You make the short but steep ascent to the large stones, turn around and...prepare to be even more breathless! On a clear night you can see the whole city from east to west, as well as almost entirely from south to north.
Pre-dusk back when Nanshan Plaza hadn't gotten to full height. Elephant mountain.
A secret(ish) spot high above Da'an District, looking northeast.
Another popular spot is up on Maokong, especially the gondola on the way up and down. From Maokong proper you're blocking out a bit of the city, but you'll get a good distance view of Taipei 101. Think post sunset tea or dinner up here, more an add on to a post zoo trip rather than a night view destination into itself.
Near Jiantan MRT is the similarly named Jiantan Mountain. The view faces to the south and it's a popular spot for watching airplanes take off and land from the nearby Songshan Airport. It's a short hike up behind the Grand Hotel which can be accessed across Zhongshan road from the MRT on a short uphill hiking path.
Jiantan has a similar view, flipped left to right as this Neihu view.
The ferris wheel at Miramar at Jiannan Road MRT was once a great place to have a ride and see the whole city. The past few years some new buildings have been completed that largely block out the best angles of the city. The wheel turns slow and it's fun to take a ride with your friends, but the value of the view has dropped significantly.
The wheel with the obviously huge buildings in sight. Taipei 101 in the distance.
DaZhi bridge, seen from Meiti Park. Sunsets are hugely popular here.
Full disclosure: I've got a few secret spots that I wouldn't share unless I take someone there myself. The ones here are the most popular and are always good. If you live in Taipei and want to get out and find some new spots, you'll be richly rewarded when you go for a night view and you're the only one there.