Tokyo Skytree (東京スカイツリー), located in Sumida Ward, Tokyo, had a shaky start when it opened in 2012, failing to meet initial visitor targets after shutdowns due to bad weather (wind!) and expensive pricing. However, in recent years it has gone from strength to strength and numbers are expected to top 30 million very soon. This doesn't include visitors to Tokyo Skytree town (Solamachi =ソラマチ) a hugely popular commercial complex at the base of the tower full of restaurants, shops, concert spaces, an aquarium, a planetarium, and (at the moment) an ice-rink! See the floor plan here.
The Skytree, at 634 metres, is currently the tallest free-standing broadcasting tower in the world and the second tallest structure, though is not on the list of tallest buildings because it fails to meet the condition of having "continuously occupiable floors." The height itself is well known by most Japanese because the numbers 6, 3, and 4 can be read "mu", "sa", and "shi" the name of the ancient province that includes modern day Tokyo. Ticket prices depend on how high you want to go: it'll cost you ¥2,060 to go up to the Tenbo Deck (340-350m) and another ¥1,030 to go all the way to the Tenbo Galleria (Tenbō Kairō=展望回廊) at between 445-450m. At the moment there is a special asa-wari (朝割) or morning discount deal for weekdays from 8:00 to 9:30am (but these need to be bought in advance).
The lifts are the fastest large capacity lifts in Japan at 600m/minute which will get you to the Tenbo Deck in 50 seconds! Be warned though that you may have to queue a while - however a fast track ticket is available (at a premium price). But if you want to save some money, there is an alternative: the observatory at Tokyo Metropolitan Government Building in Shinjuku is not quite as high but you can still see Mount Fuji on a sunny day - and best of all it's free!