Or else Windsor Castle.
April 2006. After a couple of trips to London under our belts, we got adventurous and just took the train to Windsor without pre-booking something. It turned out to be a very nice full-day visit, before returning late-afternoon to London.
In spite of being a beautiful historical site, it can be hard to get a decent shot without tourists elbowing one another everywhere. That's the problem with it being a popular royal residence. As it turned out, the Queen was in residence at the time, but of course, we didn't see her.