I would argue that the British Empire, as the precursor to the modern western world, is the greatest empire of all time. I have listed below some of the reasons for my choice.
Science and Technology
Technology - The British Empire made huge strides in technology. The pace of development was rapid and helped to furnish Britain with the resources necessary to support such a huge Empire. Britain invented, amongst other things - The Steam Engine, The Jet Engine, Combustion Engine, Lightbulb, Television, Telephone, Electrical Motor, Train, Radio, Vacuum Cleaner, World-Wide -Web. Just some of the things Britain is famed for and that have helped to shape a remould the modern world.
Science
Britain has contibuted some of the worlds most famous scientists to their respective fields: Newton, Fleming, Bell, Stephenson, Watt, Brunel.
Culture
An embarrassment of riches:
Literature: Chaucer, Shakespeare, Milton, Burns, Wordsworth, Dickens, Woolfe, Austen, Tolkien to name but a few.
Art - Godfrey Kneller, William Blake, Raeburn, John Constable, George Frederick Watts.
Architecture - Buckingham Palace, Forth Rail Bridge, Big Ben and the Houses of Parliament, Tower of London, Edinburgh Castle, Westminster Abbey; Cathedrals - Durham, Canterbury, Salisbury.
Sports (Inventors of) - Football, Cricket, Rugby, Snooker all were first played in Britain.
Healthcare - Invented Antibiotics and modern hospital/nursing environment through Flemming and Florence Nightingale.
Economics - Adam Smith's 'The Wealth of Nations' paved the way for global trading as it exists today.
Food - British cuisine enjoyed throughout te world. English Breakfast, Fish and Chips, Sandwich, Roast Beef etc.
Empire - Controlled one quarter of the worlds land mass. No other empire has ever come near the global power which Britain held. Annexed Canada, Australia, India, New Zealand, Egypt, Nigeria, South Africa, Hong Kong, Malaysia, Sudan and many small island states to the empire, (oh and of course the US) which existed, in parts, between the late sixteenth to mid twentieth centuries. This is approximately 350 to 400 years depending on when you consider it to begin and end, though, of course, it could still be argued to be in existence today.
Military - Exceptionally well drilled army and navy. The largest navy fleet ever to have existed was the British and this allowed for Britain to boss the world on the seas and to claim such large swathes of land. Tactically superior fighting skills allowed the Brits to conquer and hold land where numbers were drastically inferior.
Politics - It was (and still is) Britains political system which was the model for that of most, if not all, of the colonies under its governance. Whilst Britain undeniably created problems when withdrawing from parts of her empire and leaving them to self-governance she also helped to bring order and society to those who had never experienced it before. I'm not advocationg colonialism here, just stating that Britain did, in her time in control of these countries build the infrastructure and political basis for the continued government of many of these countries. Obviously there have been failures, but look to the good that was done also. We can see Canada, New Zealand and Australia as versions of democracy and self-governance which it could easily be argued (with more space and time than I have here unfortunately!) have surpassed modern Britain. The Indian railway network, for example, was British built and now transports 20m passengers daily over 40,000 miles of track.
I think that the British Empire was responsible for many technological, cultural, political and social advances. We live in a world today which is based around the foundations laid by the British Empire. English is the primary language of the world. The legacy of the British Empire is clear for all to see, and though Britain has now ceded most of her colonies and is in steady decline the British Empire remains, for me, the greatest in terms of size, power, innovation and influence, that has ever existed.