Arseface in Preacher, whose face was destroyed by a shotgun blast in a botched suicide attempt. 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!"
In PS238, there's an alien who becomes a student at the school. He's called 'Prospero' because no one has any idea what his real name is. His speech is represented as squiggles, and only Angie understands it. Word of God is that his speech is written in English using the Camouflage font, which is all squiggles. This doesn't help much; one of his speeches is actually a recipe.
* In the Star Wars comic Tales of the Jedi: The Golden Age of the Sith, the upper-class Sith, most of whom have at least some human ancestry (Naga Sadow barely looks like an alien at all) speak Basic (which is rendered in English for the benefit of the reader, of course). The narrator then reveals, however, that there are several castes of Sith - and the thug race, the Massassi, are the most ape-like of the species and cannot speak intelligibly at all, instead using the ancient Sith tongue, which looks and sounds to non-Sith like barbaric growling. The letterer even rendered their speech in a bizarre-looking font to emphasize how weird it is. Presumably, the crossbreed "human" Sith are bilingual and can understand them.