Arseface in Preacher. Most of his dialogue is subtitled (in a comic!), but somewhere in the middle of the story it dropped out for a while. Thankfully, a careful reader can make out pretty much all of his lines with a little sounding out of his gibberish. Apparently, the only characters who could ever understand him were Jesse and Lorrie Bobbs.
Doop, from Peter Milligan and Mike Allred's run on X-Force/X-Statix, spoke entirely in his own alien language, which other characters apparently could understand, though they would reply in English. His word bubbles were unintelligible until a reader figured out the substitution cipher used to write his dialogue.
It is later established that "Doop-speak" is an actual language (that, as a member of the team, you eventually come to learn) with its own grammar and rules, and that while Doop can easily understand English, he seems genuinely incapable of speaking anything but Doop-speak.
Eventually, we actually hear a conversation from Doop's perspective and understand him perfectly, though this isn't until Nation X.
He does prove capable of speaking English on at least one occasion, after temporarily absorbing Thor's hammer.
Mazikeen in The Sandman is a demon who is missing half of her face. Neil Gaiman apparently wrote her dialog by trying to speak using only half of his mouth and wrote down what came out phonetically. It's almost impossible to understand what exactly she's saying.
Not completely impossible if you sound out her lines slowly. Fortunately, it's pretty easy to figure out what she's saying based on context, even if you can't decode the words.
In The Great Ten, Shaolin Robot speaks solely in hexagrams from the I Ching. Thankfully, his speech is usually shown translated in a footnote - and often he only expresses one concept or idea at a time.
In the current version of Blue Beetle, the scarab communicates with Jaime using strange letters that represents an alien language that Jaime can understand because he's "linked" to it. Whether this is Wingdinglish or just random symbols seems to be Depending on the Writer (or possibly Depending On The Letterer).
Bini from Mingamanga, despite being the only native German of the four boys. Mustafa lampshades it: "And they tell me I have to learn German!"