What about a non-royal head of state and a head of government who was eventually ennobled?

FDR and Winston Churchill became very good friends during the course of the Second World War. They were in a position of rare importance and had the benefit of speaking the same language. There are quite a few quotes from Churchill describing his fondness for Roosevelt, but I'm going to be limited to what's online, since I'm at "work" and away from my books. There's also an amazing article concerning the Churchills Christmas visit to the White House written by Eleanor Roosevelt published posthumously by "The Atlantic" in 1965. She says,

My husband was not given to sitting up late at night after dinner as a rule, but during Mr. Churchill's visits he stayed up, and I am sure he was deeply interested at all times, for they seemed from the very first not only to have a good understanding of each other and an ability to work together easily, but also to enjoy each other's company. They both loved history, both loved the navy, and while I think Mr. Churchill had a more catholic interest in literature, they had some particular literary interests in common.

To elaborate on a particular point, during WWI, FDR was Assistant Secretary of the Navy and Churchill was briefly First Lord of the Admiralty. They were both inordinately fond of ships and took a very personal interest in their respective fleets, something which they bonded over easily. Churchill himself was quite descriptive as well and once said,

"Meeting Franklin Roosevelt was like opening your first bottle of champagne; knowing him was like drinking it."

He cared deeply about Roosevelt and often wanted to share things important to him which were unrelated to the war or politics. When the two men were in Morocco preparing for the Casablanca Conference, Churchill wanted to show Roosevelt the view of the Atlas Mountains from the top of a tower attached to the building they were staying in. He helped several staff officers carry Roosevelt up the stairs so he could share the view. When he received news of Roosevelt's death in April 1945, he immediately cabled Eleanor with the following message,

I have lost a dear and cherished friendship which was forged in the fire of war. I trust you may find consolation in the magnitude of his work and the glory of his name

He described the feeling as if having been struck by "a physical blow" and broke down while delivering the news to the House of Commons.

Queen Charlotte of of Great Britain & Ireland and Queen Marie-Antoinette of France kept a friendship throughout the years. As with many of the friendships Marie-Antoinette kept, the two never met face to face - it was simply not done for a Queen-Consort to leave her country, and for two of them to meet would have required some extraordinary circumstances indeed. They shared interest in music, and both had come from German speaking countries to the great courts of Europe.

Likewise, both of them suffered turbulent last days. Queen Charlotte was said to be particularly affected by the bloody events of the Revolution and the death of her friend, and she too lost her husband King George III to madness, though she pre-deceased him.

Just a short one before I head out the door:

The famed English author Rudyard Kipling counted among his friends King George V of the United Kingdom, with whom he shared a number of professional and personal interests. Both were (as you might expect) deeply concerned about the status and role of the British Empire during a time of many crises and transitions, but they also bonded over their interest in new technologies and the possibilities they afforded. Both were enthusiastic motorists at a time when cars were still expensive curiosities, and Kipling would often attempt to break his own speed records when driving to visit his royal friend wherever he happened to be staying. They were also both very interested in the power of the new wireless radio, and in 1932 Kipling authored the first Royal Christmas Address to be broadcast in that medium -- a tradition that persists today, though it has since expanded to additional and more modern formats.

Kipling and George V were both born in 1865. They experienced the many challenges facing the Empire as contemporaries, and eventually as friends. They both knew the pain of losing a son -- Kipling's only son John at the Battle of Loos in 1915, and George's youngest son, also named John, during an epileptic attack in the winter of 1919. The post-war years saw a heavy joint involvement by the pair in the work of the Imperial War Graves Commission, registering itself most publicly in a prominent pilgrimage conducted by the King, Kipling, Earl Haig and Sir Fabian Ware to many of the major war cemeteries and memorials in France and Flanders over the course of several weeks in 1922.

Rudyard Kipling died January 18th, 1936. The King followed him to the grave two days later.

Here's a fun little one. The King was close friends with the Grand Master.

Elvis Presley was friends with Ed Parker, Grand Master and founder of American Kenpo Karate. He was Presley's teacher, and eventually presented him with his black belt. He also created and hosted the International Karate Championships in Long Beach, where people including Chuck Norris and Bruce Lee competed.