Melissa Joan Hart
Samuel Langhorne Clemens
Linda Day George
Farrah Fawcett Majors
Smokey the Bear
Thomas the Tank Engine
Felix the Cat
Popeye the Sailor
Attaboy! Separate pile already started upthread, but:
Attila the Hun
Alexander the Great
Jabba the Hutt
Erik the Red
Follow the Leader! 
Antonio Carlos Jobim.
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John Wilkes Booth
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Lee Harvey Oswald
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Mark David Chapman
Rodney Alan Greenblatt
David Alan Grier
In the American South, it was very common to give both boys and girls names like Billy Joe or Ella Mae, and those would often be their formal birth certificate name, not just short for William Joseph. My mother went by Mary Ann all her life.
In the case of notoriety, people like Lee Harvey Oswald and James Earl Ray and Francis Gary Powers were given those full name treatments by the news media from official police or court documents, and most of their friends probably didn’t even know their middle names before they did something to get in the papers.
Lyndon Baines Johnson
Lady Bird Johnson
Joyce Carol Oates
…and to stretch it:
John Jacob Jingleheimer Schmidt
Wile E. Coyote
Jesus H. Christ
John Q. Public
Remember the album Frank did with “Tone” where he went by
Francis Albert Sinatra?
I may have known it before but just lately I was reacquainted with Ann-Margret Olsson as her full name.
Maybe similar names could be incorporated into this “fun and games with names” survey.
Cher
Seal
and other one-name celebs, too.
(I may have also known before but Ann-Margret is 8 months older than me! )
Kareem Abdul Jabbar
Mary Lou Retton
David Allan Coe
W.E.B. DuBois
Cecil B. DeMille
Michael J Fox
Although the late President Truman was also known as just plain Harry, he was usually referred to as Harry S Truman.
Joe Bob Briggs
He’s not best known as a double-name type. It’s extremely rare that anybody uses this.
Yo Yo Ma!
…and John Paul Jones.
(Can YOU guess which one plays bass?)
Iirc, Jobim’s nickname was Tom. The only Jobim song I’ve heard Sinatra sing (i guess it was from that album) was Dindi, and i thought it sucked balls.
Something I read, which may not be the case for each person on the list, but explains the commonness of the use of middle initials, was the fact that the number of first names just wasn’t as varied, and people were much more commonly named after a grandfather or uncle, but with a different middle name (because that grandfather might have a son of his own who was firstname middlename, junior), so Robert E Lee, for example, might be named after a grandfather who had a son who was Robert A Lee, jr.
You occasionally got some people who were distinguished by Roman numerals, if they had the exact name of a living relative, but not their own father. If John James Smith had a son named John James Smith, jr. who died in childhood, JJS’s second son, who was named Edward Robert, after his grandfathers, Edward and Robert, called Ned, but styled in writing Edward R. Smith, because his father’s older brother, Edward Samuel Smith, jr., already was Edward Smith in writing. Anyway, since there was no JJS, jr. to have a child, Ned named his first son John James Smith II.
Come to think of it, before about 1880, there were fewer surnames in the US as well. The great waves of immigration from Europe began after the Civil War.
When a family “daughtered out,” which is to say, the family name was going to be lost, because only daughters were born in a generation, even within the extended family, if the family had some social standing, usually an oldest daughter of an oldest son, who would be able to inherit, would marry someone slightly below her, but “with promise”-- maybe someone who had distinguished himself in the military, of been the first in his family to go to college, and done really well academically, and the couple would take the dual name, the names of both families, and as a couple, inherit what the family of higher standing had, including that family’s place on the social register.
In the UK, there may have been some system of transferring a title to a son-in-law, and then hyphenating a name. I’m not sure how all that works.
In the specific case of (George) Bernard Shaw, I think it was just that people called him Bernard, rather than George, who knew him well (most people called him Mr. Shaw).
It’s become common in the last couple of generations, as women who had careers before they married, did not want to change their names after marriage, and then when children were born, wanted their children to have the mother’s name as well, so the children has hyphenated names. Some couples I know, both took on the hyphenated name. I do know of a Goldman and Bruner (the latter means Browner) who decided after they were married their name would be Sienna (really; not making that up; I am good friends with one of the bridesmaids). It’s a good thing they weren’t Wasserman and Erdman, or their name would be Mud.
Tommy Lee Jones?
I believe a lot of showbiz names use middle initials (Michael J. Fox) or middle names (like the above example) because the Screen Actors Guild requires a unique name to avoid confusion among credits…so if “Michael Fox” is taken, you add an initial or a middle name – even if it’s fake.