I bet you weren't expecting that! Yes, there's an actual-size stone nose about 2 meters up the wall. Not too much is known about it but a policeman told me it was for the mounted police to touch as they go through the arch so many have suggested it is Napoleon's nose and they are 'rubbing his nose in it'. Others say it's the Duke of Wellington's nose (it was quite large, I'm told) so touching the nose is a good luck thing.
In 2011, the artist who put up this nose, and many others across London, told the Evening Standard that he did it back in 1997 as a response to all the CCTV being installed across town. So, now we have the answer! It was about being nosey.
I wasted many hours looking for the other noses put up around town by Rick Buckley but I'm pleased to say Peter Berthoud, a rather wonderful City of Westminster Guide, has devised a Seven Noses of Soho Walk which I've tried, enjoyed and can recommend highly.
In the area you can find some other unusual sights including the memorial to Giro the Nazi Dog, the World's Smallest Police Station, and a popular Harry Potter Film Location in London.