3 days in London. What do I absolutely have to see?

Let me add to what you've seen here that it would be great to simply walk along the Thames, both banks ideally. So many things are so near each other, and the atmosphere is fantastic. This is a place that grew up around a river, so walking that river will show you the place. It's ancient and modern, traditional and ever changing, and I think the riverfront shows that to you best. The dotted lines on this map show the walkways at the water:

If you start at Westminster bridge and walk the south bank until you get to Tower Bridge, then come back across the north bank, you will pass literally dozens of guidebook things. London Eye, Tate Modern, the Globe, the Tower, the Monument (to the Fire of London, so worth a climb), train stations and bridges you've heard of, and so much more. Plus things that aren't in guidebooks but are a big part of the experience for me. I love the benches along the river with the lion heads on the arms. Sometimes there are people making sandcastles down at the water just until the tide comes in. I love the juxtapositions and the unexpected delights, and I always see something unexpected.

The British Museum is free, and amazing! (If you like that sort of thing, and I do).

Agree with Mark's list above. The catch though is that you've only three days, so the key thing is to prioritize. You could easily spend weeks doing nothing but visiting Museums in London and still only scratch the surface - I'd suggest perhaps taking a look at the museum web sites in advance, and pick just one (or two, if you're a big museum buff) that best reflects your interests.

Given the short amount of time, it might be worth taking a bus tour. I usually avoid these and tend to regard them as being 'touristy', preferring to see the main sights by foot and tube on my own terms, but they are a great way of getting a good overview of the city and at least checking off the main sights while getting your bearings. Do this early on your first day, and you can then decide what parts to fill in or concentrate on on your other days. Some of these bus companies basically run a fixed route in one direction and have guided commentary, and once you buy your ticket, you can use it for the rest of the day, hopping on or off at the various stops as you want.

Also consider getting a copy of Time Out London as soon as you land, it has good listings of what's on in all the museums and theaters with reviews. It will also list what live music and clubs nights are on, so can be a useful guide for figuring out which of the city's nightlife options will be most appealing to you. I think there's also a couple of free 'event guide' brochures available, though can't remember the names of them at the moment.

Also, think carefully about how to use day time vs evening time: a museum can easily eat up a hefty chunk of daytime, and by the time you're leaving, other daytime attractions will be closing for the evening. So if you're thinking of doing the London Eye, if you do it in the evening instead, it won't eat up otherwise valuable day time.

If you're in town on a Wednesday onwards, consider going to see a show in the West End. If you really want to see a specific show, perhaps book a ticket in advance, otherwise head to the TKTS discount ticket booth at the south end of Leicester Square in the morning and see what shows are going for later that evening.

Lines for something like the Tower of London can get quite long in peak tourist times, so check to see if they have a website that lets you book in advance, or make it the first thing you do on a specific day and get there early so you avoid spending too much of your day standing in line.

Also recommend checking out a pub or two; you'll find pubs scattered all over London, Time Out has a list of some historic ones here. Pubs are usually a lot cosier than US bars are, and usually have a different selection of beers on tap than you'll find in the US. Notable are the "real ales", which are pumped into a glass via hefty manual levers rather than the taps you see at most bars.

On top of the other listed attractions I would recommend Greenwich - mostly park and the observatory - which is on the UNESCO World Heritage List.

If you have some spare time, you can link it with the nearby Canary Wharf - the new financial district - if you like modern city centres. On your way back to the centre, just before Tower Hill you have St. Catherine's Dock - a nice example of how London Docklands look like (old ports transformed into residential areas.

To be honest with you, if you have only 3 days I would skip most of the museums (even though they are very good), just go to the British Museum.

Mark's answer said it all!
The only thing I would add is to attend an Evensong service at St. Paul's, or Westminster, or St. Martin in the Fields.

They usually start at 5 (17:00) and the place will be closed then to tourists. Arrive during late afternoon, when tourist hours are still open, and just explain to the guards, staff, that you're staying for Evensong.

A wonderful way to take time to relax, reflect and hear glorious music, no matter what your religious or spiritual background.

I think Mark has covered the most of big ticket sights in his response. The V&A is one of my favourite museums and also well worth a look, and you'd be crazy not to check out the Tate Modern (after you've checked out St Paul's, you can get there by strolling across the Millennium Bridge).

Some other good museums are the Museum of London and the Sir John Soane Museum which is a tucked away gem. The Banqueting House on Whitehall is also worth a look if you're in the area (which you no doubt will be).

Lots of good markets too - Spitalfields, Sunday UpMarket at the Old Truman Brewery, Borough Market (although a bit crowded with tourists these days) etc.

Really you could go on forever, all down to individual tastes and how much time you have!

Oh and you can't go past a good walking tour - I did a few of these when I first arrived and they're great for getting the low down on the history from locals. Very knowledgeable and entertaining.