Rome. Been there a few times now and it never ceases to amaze. I studied Latin in high school and I'm a sucker for ancient history, I suppose Greece is on my wishlist too although it doesn't seem too stable these days so I'm not sure about that. Rome however is absolutely wonderful. If you've read the stories beforehand, you can just stroll along the buildings and wonder about all the great figures that must have walked there, the decision that were taken, the empire that was built. The same goes for Greece of course in a way, having laid the foundation for later European powers. Especially if you believe the story of Aeneas of Troy.
I guess similarly I really enjoy old Chinese sites but unfortunately they don't have that ability to transport you because of the city sprawl around and through it. I think Rome manages to keep that charm more because the city itself just evolved differently. Walking through Beijing it is hard to forget the modern world around you, even if you are in the Temple of Heaven, Forbidden City. I think the Summer Palace works in a way, because you can find a spot to be alone for a bit and think about it. Of course I can also attribute this to the fact that I know considerably more about Roman history than Chinese history. Personally I enjoy knowing certain small specific things, such as the emperor hanging himself from a tree on Coal Hill and then sitting around the place and wondering about it.
Perhaps as an extension to this I would really love to visit the archeological site of Troy, but the way I've built it up in my head it would probably be disappointing. Especially given the way they excavated the area with Schliemann messing some stuff up early on. I don't know, perhaps I should leave it to the imagination.