Corsica, Ethiopia, Mexico, Uruguay, Sarawak, Katanga and Morocco all asked to be annexed by the British Empire but were refused

According to Robert Tombs' bestselling and quite wonderful book The English and their History, Corsica, Ethiopia, Mexico, Uruguay, Sarawak, Katanga and Morocco all asked to be annexed by the British Empire but were refused.

Robert Tombs supervised me at the university when the world was young and Mrs. Thatcher had just led her party to a second victory. 

I can't recommend Dr. Tombs' book too highly. He has written a wonderful Tory (i.e. accurate and non political) history, free not of all liberal prejudices but of many, very insightful and full of brilliance. He spent his career as a historian of France not England.

He explains how the Empire came to exist, despite the forebodings of Liberals like Cobden. (Another Liberal, not quoted by Dr Tombs, Hilaire Belloc said the British Empire was an ugly name for an ugly thing.) 

Dr. Tombs:

Occupying new territories usually turned on some combination of strategic
interests, local responses, economic importance, lobbying by interested parties and the wish to keep other European states out.

Indian princely states and many other places were taken over because they were failed states. 

The Empire cost Britain a lot of money and many hoped that it would evolve into an association of independent countries.

The British had no doubt (like George W. Bush invading Iraq) that they had a civilising mission. Unlike Bush the Younger, the British did have. In particular the British genuinely sought to stamp out slavery long before the Americans got in on the act. Castlereagh persuaded the great powers (who were reluctant) to condemn slavery in the Treaty of Vienna in 1815, the first time there was a human rights clause in a major treaty.

The British desire to abolish slavery around the world meant patrolling the seas looking for slavers. It was expensive and gruelling for the sailors. The British policy made the British very unpopular with African rulers, who were keen to keep slavery even if they could no longer, thanks to the Royal Navy, sell slaves abroad. Britain annexed Nigeria, for example, because the ruler refused to abolish slavery.

In the end, as a result of the Royal Navy's efforts, ownership of slaves by white men was stamped out, although Arabs, blacks, Chinese and others continued to own slaves. The British sailors, soldiers and statesmen who achieved this are now vilified in some parts of the Academe as dead, white males. 

Innumerable other benefits came from colonialism, the first of them being order. Others included conversion to Christianity, roads and medicines, European literature, cricket, fair trials, French wine and much else. Yet nowadays there is a tendency to think of imperialism as something bad, even another thought crime.

In Romania the idea is universal that the colonial empires were wicked and greedy rackets. Romanians were lucky to have escaped lightly under Ottoman suzerainty compared with their neighbours, but Turkish indirect rule and Hungarian and Austrian direct rule rankle deeply.