London

Oh London...
It's an amazing city full of life, people and things to do. I spent 10 days there and I know for a fact that even if I spent 10 years there, I still wouldn't manage to see everything London has to offer. Here's just a sample of what I did!
Sights
The Big Ben, the Westminster Abbey, the London Eye, Buckingham Palace, the Tower of London and the tower Bridge... all these world-famous landmarks are all in one city. It seems almost impossible for one single city to have so many completely different world-famous sights, but London definitely does, and it almost makes you feel like you're travelling in more than one city as you try to see them all (hopefully not in one day).

Changing of the Guard at Buckingham Palace

Museums
London has a gazillions museums of all types, and the majority of them are free, so it was a pretty good (and cheap) passtime. The British Museum is an amazing place to start. It has over 6 floors and talks about culture and history on every continent, so you're sure to find something interest, no matter your interests.
I absolutely loved going to the Shakespeare's The Globe Theatre. There's a museum about the history of theatre in London, but also of course, about Shakespeare. The Globe was rebuilt to look as historically precise as possible, and let me tell you, it's definitely impressive. I wish I had been there during the summer to see a live play and stand in front of the stage like a "Groundling"!
You could also spend hours and hours at the Victoria and Albert Museum. It's the biggest art and design museum in the world! It has an amazing collection of all sorts of objects, jewelry, statues and paintings.
If it's history you like, make sure to go to The Royal Greenwich Observatory, and go stand on the Prime Meridian! There's a whole museum attached to it about time-keeping, watches and how we calculate time in our modern world.

The Quirky side of London
Little places like Carnaby Street, a small shopping area in Soho, or the Covent Garden area which is another small market, or taking a Jack the Ripper walking tour at night, go to Abbey Road and take a picture like the Beatles, walk around Picadilly Circus and watching street performers, check out the M&M Store or Madame Tussauds,

Magical London
For any Harry Potter fan, London is an obvious destination. You can go to King's Cross Station and take a picture at the Platform 9 3/4, you can do a free walking tour of the film locations, you can walk on the Millenium Bridge, which gets destroyed by Death Eaters in the movie, and of course, you can go on a Studio Tour (That will be a whole post by itself, believe me!)

As you can see, I breezed over 10 days of visting. It was a whirlwind of a city and though it's much much too big for me, it was definitely worth it.