In 1914 the Royal Navy was more powerful than any other two navies combined. It was the means through which the British Empire projected power around the globe, and had a presence in literally every ocean on Earth. I once heard historian Jed Martin joke that the definition of an island at this time was 'not a body of land surrounded by water, but a body of land surrounded by the Royal Navy'. The power of the Empire was based on the bluff that made it appear that, whoever or wherever you were, if they had to, the Royal Navy could attack you. In practice, the Navy had provided ample proof for this myth by successfully ending the Atlantic Slave Trade in the 19th century, and by sending gunboats into Chinese ports during the Opium Wars to open up China to western trade.

At this time the Empire also controlled perhaps 25% of the world (that's the pink bits on the old maps of Empire) and also had significant financial interests in and financial control over much of South America and China (historians such as Robinson and Gallagher sometimes call this 'informal Empire' to contrast with the 'formal Empire' of de jure control).

This 'informal Empire' of economic ties has also been called 'Imperialism of Free Trade', which saw Britain maintain global interests under the pretext of guaranteeing free trade through the Royal Navy as an effective world police force. (In principle, they did guarantee free trade as a basic common good. In practice, the effect of this was to shore up British interests as they benefited from the removal of tariff barriers in other countries). Although the total amount of actual territory controlled by the Empire peaked after the war, the Informal Empire of business contacts and economic control arguably peaked beforehand.

The First World War was so massively expensive to the belligerents that they would never fully recover financially. Following financial crises in the early '20s and again after 1929, it took another world war to kick-start industry in Britain, Europe and America. Britain was financially in hock to the Americans at this stage, and relied on US finance to pay for its Empire, its Armies, and its native industry. The disruption to global trade caused by the War also saw many countries seek alternatives to Indian cotton, British coal and British shipbuilding, which had been bulwarks of the Imperial economy.

The interwar years saw definitive independence granted to the colonies of settlement (Canada, Australia, New Zealand, et al.) through the 1931 Statute of Westminster, which broke the legislative link between the London Parliament and Parliaments in the Colonies. Although the King remained the Head of State, these countries could now make their own laws with no further recourse to the UK.

The Second World War finished the Empire as the world's leading power. Specifically, the sinking on 10 Dec 1941 of the HMS Prince of Wales and the HMS Repulse off Singapore completely ended the British Naval presence in East Asia, and also ended the myth that the Royal Navy could defend British interests any time, any where. The British Economy was broken by the high cost of the war effort, and, following the war, British rule in India was wound up in two short years, and followed over the next two decades by independence for most of the remaining Empire.

Assessing all this information, I would argue that the peak of the Empire was 1914. Indeed, that the very peak was the summer before war was declared. Although Germany and the United States were catching up economically, the British Empire remained the dominant economic force and the main player in Global Trade. The Victorian 'Imperialism of Free Trade' model had opened up the entire world to British trade, and British interests held sway on every continent. This model was guaranteed by the Royal Navy, which was much more powerful than any other two navies combined, and would remain so until the US underwent massive re-armament during WWII.

This is a subjective assessment of information available to me at this time and I'd very much appreciate any and all alternative theories.

Sources: Robinson and Gallagher, 'The Imperialism of Free Trade'

Martin, Jed, 'Was there a British Empire'

Stockwell, Sarah, 'Ends of Empire' in The British Empire: Themes and Perspectives

Edit: extra specificity to summer 1914.

Edit: PS: A fairly accurate, if obviously dramatised, rendering of what British Culture looked like in this last summer before the war can be gleaned from Season 1 of Downton Abbey. The creators do place a high value on accurately representing life in an English country house, and though I wouldn't recommend referencing it in any journals, it would be fantastic for a first look at Britain during this time.

Edit: paragraphing.

Edit: to answer your question: no, there isn't a generally agreed upon year for the peak of Empire. 1956 (Suez Crisis) 1968 (withdrawal from East of Suez), 1997 (Hong Kong handover) and other years have been suggested for the End of the Empire. The peak of Empire does not admit of such neat theories but the one I have assembled above carries some currency based on the available sources.

1914 is a good candidate, but I suggest 1896, Queen Victoria's Diamond Jubilee, as the true peak of British power. Germany had not yet begun its naval challenge; it's true that the Royal Navy was larger (and way more powerful in absolute terms) in 1914, but it's also true that by 1914 the two-power standard had effectively been given up and the real standard was "Germany plus a bit". Note the withdrawals of capital ships from the Mediterranean and Asian stations between 1905 and 1914, to match the Hochseeflotte; note also the naval agreement with France, allowing Britain to concentrate on the Channel while France defended the Med. These are not the actions of an expanding power.

1896 is also a few years before Britain spent three years, 40000 casualties, and 200 million pounds (and invented the concentration camp) to subdue two poxy colonial republics with a total population rather less than half a million.

Britain in 1914 was more powerful in absolute terms than it had been in 1896, measured in GDP, total throw weight of the Royal Navy, number of hulls, rifle strength and fighting experience of the army, or practically any other metric you choose. But in relative terms it had rather gone backwards; Germany had grown faster. The Royal Navy was larger because it needed to be larger; in 1896 there was no effective challenger anywhere on the oceans; in 1914 the Hochseeflotte positively looked forward to Der Tag. (That the only thing remotely approaching a real fight between the two navies consisted of the German fleet desperately trying to skedaddle and the British trying to bring their guns to bear is a separate matter.) The Britain of 1914 was a mobilised giant, anxiously looking across the water at the challenger, and putting every penny it could find into defense. In 1896 there was no question of who was first among the Powers, and Britain didn't spend as much on battleships because it could afford not to.

The height of power for the British was almost certainly during the long peace between the defeat of Napoleon and WWI (I also lean towards pre-Boer War, but I think this is splitting hairs. Either time is an excellent argument). However, to add a little spice to the mix I am going to bring up a different time. Let's consider the era just after the Seven Year's War. The French had been defeated in both Europe and the Colonies and were forced to hand over both treasure and land, expanding the British Empire and stuffing its coffers (well, not really. The British government were terribly in debt and began to levy some very unpopular taxes). The Spanish and Portuguese Empires were clearly in decline and the German Empire was not yet united.

The British had clear control east of the Mississippi River and Canada, then the richest parts of North America. India was firmly in the hands of the British and it looked as though no one could stand in the way of British Imperial ambitions. They had a powerful navy and an army with enough competence to beat the French. Until the conflict with North American colonists the British were clearly on top of the European power heap.