If I had to pick the most iconic and original piece of architecture in the world, the one I knew from a very young age, it would have to be the Sydney Opera House. A beautiful structure reminiscent of sails, and beautiful from afar, up close, and spectacular when lit up at night -- one day I might even climb it. When I first saw it at the age of 11, it was a real treat, something so awe-inspiring and special it made a memorable impact.
I think the era it was built (started in 1959 and finished in 1973) and the fact that it remains so iconic makes it legendary. Few recently man made structures in the world have had such a long-lasting effect other than the Eiffel Tower (#898) in Paris. It was designed by Jørn Utzon, and when construction commenced, many geometrical and structural issues with the design remained, not to mention funding and political support. We can only be glad that they all were resolved and the construction succeeded!