Sixpacksaid once that he never changed his costume in all his adventures.
DC’s The Creeper doesn’t seem to have changed.
1968 debut version
2007 version
Of course, if he dabbled with some alternate duds in the interim, I couldn’t say.
The recent cartoons reflect the character’s original comic-book appearance, as Mr. Mxyztplk (note the slightly different spelling).
That version was later retconned as the Earth-2 version of Mxyzptlk. Not technically the same character as the bronze-age orange-tunic version, who also appeared in animation in the 1970s/80s Super Friends toons.
Judge Dredd.
Has the Penguin always had the tux and spats?
I don’t know that I’ve ever seen him in a tuxedo, though I suppose it’s not unlikely. His traditional outfit always used to be formal morning dress (or occasionally evening dress) – which I’d have said was more of a style of dress than a costume per se. In some recent appearances he’s had a more dishevelled look, with a long overcoat, obviously influenced by his portrayal by Danny Devito in Batman Returns.
A lot of these guys never changed their original costume. It was just drawn differently in the comic books.
The Punisher?
The skull’s eyes change according to who is drawing it, does that disqualify him?
Plastic Man has had a couple of changes. Originally, the shirt was red on the right half (no sleeve) and black (with one long sleeve) on the right. Eventually it was red with sleeves. In his most recent appearance, he had red leggings, which he usually didn’t have in the past unless the colorist made a mistake.
Spectre: Jim Corrigan and Hal Jordan had significantly different costumes from each other; Hal Jordan’s had some Green Lantern touches to it.
I don’t think Wildcat’s costume ever changed significantly, although different artists sure interpreted it differently.
Even ignoring the weird stuff like Frankencastle, his Cap costume, and the whole weird racelift thing, Punisher’s had multiple distinct costumes.
In his first appearance, he wore standard superhero style tights. Now he’s typically in jeans and a t-shirt. The black and white colour scheme and skull motif are consistent, but that’s it.
The Phantom, maybe?
Whatever. If it matters to you enough to rationalize it this way, okay. Maybe next you can explain the length of Batman’s ears.
I’m not sure a gag costume used for a single storyline should count against him 
Punisher’s “costume” has varied between having white gloves and boots or not. Also, sometimes he’s just wearing a shirt with the skull on it, sometimes he’s wearing what appear to be tights.
Oh, good one! They altered it for a recent TV show, but it hasn’t changed in the comic strip in, like, 80 years.
John Romita designed him with the skull going up over his shoulders, and they toned that down by the time of his first appearance. The white boots and gloves stayed until Bill Reinhold started making them black (which kind of made more sense). The MAX imprint gave him jeans and an overcoat; Purists claim this is a separate character, but the look has leaked into the 616 books. But he started off looking like a John Romita superhero and has gradually drifted away from that look.
“rationalize”, huh? I don’t see it s a change.
Batman’s ears (and the length of that cape) are something else entirely.
I think we have a winner!
Compare this first comic:
to this recent image:
https://upload.wikimedia.org/wikipedia/en/thumb/4/46/Phantomcomics2.jpg/250px-Phantomcomics2.jpg
As an added plus, I just discovered that the Ghost Who Walks was the first superhero to wear the skintight costume that became de rigueur for all subsequent superheroes.
How about The Shadow? I believe the pulp heroes are going to win this contest.