Eat both quality French and Italian food there.
They still have real CD shops, if that matters to you.
Spend some time in the underground/subway city parts, maybe Shinjuku station, a few others.
Not sure if the old fish market is still up and running, worth a visit if it is.
Get your iPhone ready for translate functions, print and voice.
Getting lost there is great, don’t obsess over sights.
National Museum. The Western museums are decent but also not essential.
Look for a neighborhood with immigrants.
Sample Tokyo at all possible hours, if you can.
Kinokuniya bookstore is quite good. Overall I don’t love the Roppongi part of town, though, fancy bars and restaurants for expats, though fun in its own way.
Visit a Japanese working class district, such as Ikebukuro, also a major subway stop.
Look for vending machines and collections of vending machines.
The arcades there, including for children, are pretty amazing.
Try Pachinko once.
Addendum: Here are the suggestions from Scott’s readers.