On a global level I would have to say the Taj Mahal takes some beating for sheer beauty and atmosphere.
At a more local level, from Greenwich in London, the Flamsteed Observatory in Greenwich Park, high up on a hill, overlooking the Thames and all the various other buildings of note such as the Naval College, the Royal Seaman's Hospital and the National Maritime Museum rather captured my fascination as a girl.
Now named the Royal Observatory Greenwich it played a major role in the history of astronomy and navigation, and is best known for the fact that the Prime Meridian of the world (0 degrees Longitude) passes through it, and thereby gave its name to Greenwich Mean Time.
commissioned in 1675 by Charles II and designed by Sir Christopher Wren (also designed St Paul's Cathedral). King Charles also created the position of Astronomer Royal At that time the king also created the position of Astronomer Royal, the first being John Flamsteed. It is still a working observatory to this day.
Greenwich - The Prime Meridian of the World.
In 1833, to help mariners at the port of London and others in line of sight of the observatory to synchronise their clocks to GMT, Astronomer Royal John Pond installed a time ball that drops precisely at 1 p.m. (13:00) every day atop the observatory. Initially it was dropped by an operator; from 1852 it was released automatically via an electric impulse from the Shepherd Master Clock. The ball is still dropped daily at 13:00 (GMT in winter, BST in summer).
Royal Observatory Greenwich
The Shepherd Gate Clock. Don't be fooled by the appearance, this is one of the most accurate clocks in the world.