Roppongi Hills is almost a city in itself. There are places to shop, eat, live, and play. It's also a hub for high-end culture in Japan with a number of luxury restaurants, international fashion brands, and apartment towers known for housing Japanese celebrities.
1. Mori Art Museum
Asia has some great modern art museums as it is, but the Mori suddenly showed up and made a splash. Situated near the top of the Mori Tower (the big tower of the complex, lit up in blue in the photo there), it's a beautifully-designed museum with a great rotation of exhibits.
2. Tokyo City View
The soon-to-open Tokyo Sky Tree may quickly take away the title of "best city lookout point" from the top of the Mori Tower, 52 stories up, but for the time being this is the one you want. It's basically part of the art museum, so you can get a combined ticket and do exhibits and the awesome view. If it's light outside and the weather's OK, you can go outside on the roof and view the endless buildings of Tokyo al fresco.