If you were in Edinburgh for three days in the summer....

[Alan H]

...what would you do?

Besides visit the castle, of course.

[Tartan Hiker]

Well...we were there in '06 and spent an entire day in the castle and immediate vicinity, a day touring the Museum of Scotland and Holyrood, and a day just poking around the old city (shops, churches, lots of places to see and do). Walked everywhere we wanted to see from our hostel which was located about 5 blocks from the castle.

[Tartan Hiker]

At night, the old part of town gets pretty quiet (except for a few pubs and fish 'n chips shops), so
we walked across the bridge to the "new town" where the shops and night life were all wide open
until late.

[LANCER1562]

Depending on the time of year, attend the Tattoo in the Summer or if I'm there in December, I would attend the various Hogmanay celebrations. I would, of course, check out the museums, some of the old churches, maybe hire a car and check out some of the famous battle sites if they're not too far.

[chasem]

I enjoyed visiting that other castle and heading up the short hike up Arthur's Seat.

[The Wizard of BC]

You may be able to make it all the way through the museum in three days. As long as you don't mind missing a few things.

I could also easily spend three days sitting at a table of a sidewalk cafe and pointing and laughing at everyone who walks by in pants.

[cessna152towser]

Well from the perspective of a Scot who lives only 55 miles from Edinburgh, I would plan trips away from Edinburgh for each of the three days. Edinburgh is a touristy city and you aren't going to see the real Scotland or have the opportunity of meeting Scots people (other than those who interface with tourists for a living).
One day I would visit Stirling, see the castle and the old royal palace and also visit the Wallace Monument. If your wife gets bored they have a really good all-covered shopping mall in Stirling town centre.
Day two I would visit Glasgow with its historic buildings, fine architecture, beautiful parks and vibrant waterfront. Have lunch in one of the riverside pubs, sit outside on the promenade if the weather is fine. Then there's great shopping for your wife, St. Enoch Mall, Princes Gardens Mall and the Buchanan Galleries.
For the third or final day you could travel south into either the Scottish Borders or Dumfries and Galloway regions, see the gently rolling hills of the southern uplands in a landscape dotted with old tower houses and castles which bear testimony to the era of the Border Reivers. Dumfries is a nice town with a Burns Museum and you could visit the house where Burns lived in his final years and have a pint in the four hundred year old Globe Inn where Burns was a regular customer. Again Dumfries offers good shopping for your wife, with the modern Loreburn Centre just a stone's throw away from the Globe Inn, not forgetting Anderson's kilt shop where you can often find something a little bit different.

[Padraicog]

Catch the train to Glasgow every day

[KenB]

All really nice suggestions. But, if time allows, take a tour of Mary King Close (underground Edinburgh) and a Scottish Writers Tour that begins at a pub in the Grass Market and meanders through the old city, going from pub to pub, to finish at ,now wait for it..... another pub. Both are interactive tours and very entertaining. However, don't deprive yourself of seeing Holyrood, Scottish Parliament and a stroll up to the Castle.

[A Hay]

If you and your wife would be interested, the former Royal Yacht, the Britannia, is located just east of the main town. It is possible to catch a bus over to it and back. My wife, daughter, and I spent a nice couple of hours there one afternoon.
Another interesting place is the Scottish National Museum which is just west of the Royal Mile. From the rooftop viewing platform, you get a great view of the City and beyond.

Tom