People with an accent entirely their own...

My 10 year old daughter. Native speaker of three languages, she has her own damn accent in all three, which nobody can ever place. It’s the cutest thing. I think it came about due to the fact that she has absorbed different accents for different words/expressions, due to not living in a native environment for any of the languages.

And I guess Stephen Hawking has the exact opposite of an accent.

Me. One part New Orleans drawl of varying strength, one part Chicago nasal, and a tendency to not properly enunciate when excited.

Had one guy ask if I’m Canadian.

Arlen Specter was born in Wichita and grew up in Russell KS (same hometown as Bob Dole). So maybe it was that Kansas twang coming through.
Not speaking (as far as I know) but singing - what is up with Colin Meloy of the Decemberists? The first time I heard the Decemberists I spent half the album trying to place his accent, figuring it was some kind of British… Finally looked it up on Wikipedia. Nope, he grew up in Montana.

I realize his voice was exaggerated by a generation or two of impressionists, but John Wayne always struck me as having an odd way of speaking.

Humphrey Bogart…speech impediment nonwithstanding

USMNT GK Brad Freidal

All of these people, and we could throw in many more from that generation ( like Grace Kelly ), utilized variants of a Mid-Atlantic accent, a deliberately affected and taught ( back when teaching proper diction in classrooms was a thing ) social-class accent now approaching extinction. It was once the province of the wealthy, private-school attending New Englanders/Mid-Atlanticers hence its association with the posh and eventually the prissy and pretentious. Since it isn’t taught anymore and was class-specific rather than geographic it is has been declining for decades except as a put-on for comedy and the like.

The most unusual ( and very prettily musical ) accent I ever heard was my one of my history professors who was an Argentinian who had learned English in college in Britain. To my ear the combination was very subtly exotic.

Howard Cosell?

Someone on Reddit was able to discover that Tommy Wiseau grew up in Poland with the surname Wieczorkiewicz, and lived in Strasbourg illegally for a while under the name Pierre Wiseau, before moving to his aunt and uncle’s house in New Orleans. So I guess his accent is a mixture of Polish, French, and Nawlins.

WC Fields?

People may not think of the silent-film star Buster Keaton for a question like this. But he appeared in a handful of talkies in the 1930s, and he sounds weird as hell. He was born in Kansas and traveled around doing vaudeville with his parents.

Leopold Stokowski, the famous orchestra conductor. Born and grew up in England, but listen to him with Micky Mouse in the Disney film Fantasia; vaguely Eastern-European, and apparently it’s intensity could vary.