In the centre of London is a small area around one square mile in size called the 'City Of London'. This is where most of the financial services sector of London lives. It seems to have a bizarre and fascinating history. From Wikipedia:

The City of London has been around for a long time, and is mentioned in the Magna Carta:

The rights and privileges of the City of London are enshrined in Magna Carta’s clause 9—as enumerated in 1297—and, along with clauses 1 and 29, it remains in statute.
(e.g. Magna Carta states that "the City of London shall have/enjoy its ancient liberties")

Businesses can vote in the City of London elections:

.. the non-residential vote (or business vote), abolished in the rest of the country in 1969,  became an increasingly large part of the electorate.  The  non-residential vote system used disfavoured incorporated companies.   The City of London (Ward Elections) Act 2002 greatly increased the business franchise, allowing many more businesses  to be represented. In 2009, the business vote was about 24,000, greatly  exceeding residential voters.[

The City has its own Mayor, called the Lord Mayor of London. This office has existed since 1189 (!):

The office of Mayor was instituted in 1189, the first holder of the office being Henry Fitz-Ailwin de Londonestone.  The Mayor of the City of London has been elected by the City, rather  than appointed by the Sovereign, ever since a Royal Charter providing  for a Mayor was issued by King John in 1215. The title "Lord Mayor" came to be used after 1354, when it was  granted to Thomas Legge (then serving his second of two terms) by King Edward III.