For years, Taipei 101 was the tallest skyscrapers in the world. As an iconic landmark and one of the tallest buildings worldwide, the observation deck of Taipei 101 rightly becomes a must-visit. It is certainly worth seeing. To me, the most amazing part is the structure design and this 660-tonne tuned mass damper that helps the building withstand typhoons and earthquakes. Bear in mind that this 509 meter / 1669 feet building is situated on the circum-Pacific seismic belt.
However, if you want to see the view of the Taipei City with Taipei 101 in it, the Elephant Mountain Trail, or the Xiangshan Trail, is one of the best places to go. I highly recommend this mini hiking trip for those visiting Taipei 101 and the Xinyi District (信義區). Note that this is for people who do not mind lots of walking and stairs.
[Updated August 2018]