In Roppongi, you can find Tokyo Tower. It is a 333 meter (1,093 feet) tower that is open to the public to see views of Tokyo during the day and night. The top deck is 250 meters from the ground.
For 1200 yen (~$10 USD) you can head to the main observation deck near the middle of the tower, and for 3000 yen (~$25 USD) you can head closer to the top.
The outside is lit up at night, and it really stands out in the landscape.
Inside the base of the tower are several souvenir shops (mostly snack foods and anime-themed merchandise.) It's a good way to pass the time as you have to wait 15-30 minutes for your turn to go up the elevators to the upper floors.
An interesting note is that when descending from the middle of the tower, there's an option to go down the stairs, which is said to take about eight minutes to descend. There is no option to do this from the top floor.
Aside form Sky Tree, which you can see in the background and that is much higher at around 450 meters, it is the second tallest building you'll find in the Tokyo Skyline.
There's a 360-degree view of the city from the top of the tower. One of my favorite views above shows a major expressway forming a star-shaped pattern. I really wish the photos did it more justice, it's a lot more visually impressive to watch this in person.
It's definitely worth a visit if you're in the area. You're also likely to see a lot of very expensive cars driving around in Roppongi if that's your thing. For nostalgic food, you can find Hard Rock Cafe, Outback Steakhouse, and a really authentic New York-style pizzeria. I also found the Dragon Quest-themed Luida's Bar nearby.