Every single character in Banjo-Kazooie and any other sequel. All of them speak in unique sounds that need to be translated in subtitles for the player to understand.
Crash Bandicoot speaks in cheerful gibberish. The other Bandicoots and Aku Aku can understand him, but other characters' mileage varies.
Darth Nihilus from Knights of the Old Republic II communicates only in a series of tones which, it is implied, are generated telepathically. The other characters, including the player character and his apprentice Visas, seem able to understand him, but decline to translate for the benefit of the player. Nihilus later appeared in issue 5 of the comic series Star Wars: Legacy, where his only line was rendered as incomprehensible symbols that no other character bothered to translate.
Half-Life 2 has Dog, a ten-foot Robot Buddy with the build of a gorilla, who communicates entirely through expressive beeps and posture. Naturally, Wrench Wench Alyx Vance understands Dog perfectly.
Combine Soldiers are occasionally guilty of this, however they seem to have no trouble at all understanding each other. Normally, if you listen carefully you can tell exactly what they are saying, but there are a few instances where no matter how many times you hear it, it's still unintelligible:
Prison Soldier Radio: Negative, no sector containment, no conformation on * mumble* , we have secondary perimeter overrun * mumble* .
Prison Soldier Radio: Drop reinforcement teams deploy and prosecute sector Delta 7, contact confirmed with Primary Target. Target is * mumble* scan for possible * mumble* .
Snott on Earthworm Jim. He speaks in slurps, and everyone understands him. At one point, one of the characters calls him "incredibly eloquent," but this may have been sarcasm.
The Pyro class from Team Fortress 2 wears a gas mask and speaks in an unintelligible muffled voice - one can occasionally make out what he is trying to say, such as when he calls out for a medic. This is spoofed here and here.
Mocked by virtually all the classes when they dominate him. Some gems:
Scout: Repeat after me: "mmmmh mmmmh mmmmh I'm dead!"
Sniper: You are a creepy mute little bugger, ain't ya?
Q in Street Fighter III: Third Strike communicates only in odd groans and grunts. Whether he's being silenced by his steel mask or just can't talk intelligibly is unknown.
Also in the same game, we have Twelve, whose winquotes are all in binary code, therefore implying that he speaks in binary.
And then there was Blanka in Street Fighter Alpha 3, in which all he could do is grunt (even in his winquotes!). We later learn that, in the gap between SFA3 and SFII, Dan taught him to speak properly.
Torr from Fallout 2, due very limited vocabulary. However, if you play a retard (Intelligence below 4), you can have a sesquipedalian conversation with this fellow!
You get to play The Unintelligible if your intelligence is too low. There are like two people in the whole game who you can really talk to.
The clones of Gary in Fallout 3, which are the sole human inhabitants of Vault 108 by the time of the player character's arrival. For the most part, they speak only their name "Gary" in various tones (however, they can be found speaking minor NPC to NPC dialogue occasionaly).
In a Fallout: New Vegas expansion, Doctor 8 is this. With high enough Perception or Science though, you can determine his speech pattern and determine (by your speech options) what he's saying.
Some of the options following any particular bit of 8's speech are contradictary. It would work if you were to follow the conversation, but seeing the options makes it seem like the character is just making it up as he goes.
Sir Daniel Fortesque of Medievil fame, having been dead for a hundred years and reanimated as an armor-clad skeleton, has lost his lower jaw and can only mumble his way through the game.
The Legend of Zelda fanmade flashtoon parodied the concept. Since the character is a Heroic Mime but, in many modern games, can actually be heard to make noises when jumping or attacking, he encounters some difficulty in attempting to make a purchase at a store. "Hya!" "... I'm sorry?" "Hya! Hup! Hya!" "That's a really thick accent you have, there..."
In The Legend of Zelda: The Minish Cap, the Minish can only say the word "Picori" (as well as various syllables of said word)...at least until you eat the Jabber Nut, at which point you can understand them.
They talk backward in the Japanese and the German version.
And of course in The Legend of Zelda: Twilight Princess, Midna's lines are all dubbed as unintelligible gibberish, (though they appear 'translated' on screen) although apparently Link understands her just fine.
Most beings in Loco Roco speaks only in various sounds and songs. The only thing you can make out is when they spot an enemy moja, to which most loco's will say "moja! moja!"
If you listen closely you can also hear them say "Toge! Toge!" if you get too close to spikes, greet MuiMui, and even exclaim "There!" if you are close to a breakable wall.
In Lost Souls MUD, a mental disorder, productive aphasia, converts player characters and NPCs alike into this. Hilarity ensues.
One of the "silly" emotes for the female blood elf in World of Warcraft relates a story about getting a facial in the Undercity. When the storyteller protests getting a facial from someone who doesn't have a lower jaw, "She got mad. At least, I think she did. Have you heard someone trying to talk without a lower jaw?! 'RAURGHAURGHAURGHAURGH!' She sounded like a Murloc!"
Starting with Yoshi's Story, Yoshi speaks in an unintelligible language. He is voiced by one of Nintendo's sound effects men, Kazumi Totaka.
Although Yoshi has shown the ability to say a few words (aside from "Yoshi," of course, he can say "yum," "yay," "wow," etc.), in the recent Mario Kart Wii, he can be heard saying "Hello" and "Here I go!" It was rather... jarring.
And of course, he says the name of the company, Nintendo, when you first turn on the console for Yoshi's Story as well as in the Yoshi's Island GBA version, although it sounds more like he's saying "Weepinbell."
Super Mario RPG uses text, but also has it so that the Yoshi is the only one Mario can actually speak with (presumably because he speaks Mario's language). The other Yoshis can only be understood if Mario is riding the Yoshi, and having him translate.
In many games, Donkey Kong only lets out gorilla grunts.
The Yellow Devil from the original Mega Man series would like to have a word with you. Unfortunately, as one of the page quotes above indicates, there's only one.
Humorously, Cut Man tries to communicate with the Yellow Devil before the boss fight by nervously imitating its speech pattern. Judging by the Yellow Devil's response, Cut Man either wasn't convincing, or accidentally said something insulting.
Ice Man understands what the Yellow Devil is saying, but doesn't try to reply; instead, he just says "...I understand." followed by a translation. The Yellow Devil, however, doesn't seem to understand English at all, as its response here doesn't differ from what it says to the other robots.
Zombies in Urban Dead can only groan variations on "Graagh" and a vague approximation of "Brains" ("Brnhr.") unless they purchase an ability, and even so they have a loose grasp of pronounciation.
BARHAH, MAH ZAMBAH BRAZZAH!!
As Urban Dead is an MMO, players have done everything from make entire cryptographic translations of the available letters, to the more generally-used form of zombie pidgin which is exemplified above. "Barhah" is the zombie rallying cry, akin to "Sonno-joi", "Allahu Akbar", or "U-S-A! U-S-A!".
Call of Duty 4: Modern Warfare, multiplayer modes. American Marines and British SAS call out easily comprehensible things like "Changing mag!" or "Grenade!" or "Poppin' smoke!" Russian Spetznaz and Arab OpFor, on the other hand, fluctuate between speaking English ("Enemy UAV is airborne"), speaking comprehensible Russian (something that sounds like "PROSEI GRENADO!") and saying stuff which is less than obvious (something about Neutrogena, apparently, at least once in every game).
This also happens, naturally, in Call of Duty: World at War, which is based off of Call of Duty 4.
Probably only me who thinks this, but I find it completely incomprehensible what one of the marines shouts in the opening cutscene that sounds like "For Neilion!"
In Grand Theft Auto IV, even the characters themselves have difficulty understanding Little Jacob's rasta accent. One of Elizabeta's missions can be accessed when she calls Niko while arguing with Jacob, in factquote Shut the fuck up, Jacob! Niko, I hear you can understand this rasta shit! Get over here before I run out of patience and start shooting!And then you have his boss, Badman. He's so hard to understand, Jacob has to "translate" what he says.
A number of the monster classes in Disgaea 4 have at least one personality type that doesn't communicate in an understandable fashion, such as the barking Mystic Beast, crowing Cockatrice, and Bone Dragon who only says "bone". However, conversing with them in the hub gives a translation of what they're saying (Which can be rather deep for, say, a single "woof").
Who can forget Kirby, whose entire vocabulary basically consists of "Hiiii!" "Whoa!" and variations of "poyo!"?
Umaro, an optional party member in Final Fantasy VI, is a sasquatch who communicates only in grunts.
In the Wii Punch-Out!!, King Hippo has only animalistic roars.
Blazblue's Arakune quite literally has no mouth, so he fashioned himself a mask that would help him communicate with others. However, even with it Hi. wor.. sti.. ..nd t. ge. cu. off .
Most demons in Persona speak intelligibly. Grimies is not one of those demons. Its favored method of communication is strings of seemingly-random numbers. Good luck negotiating with that. Made even more fun because its questions are impossible to decipher, on top of it being a Demonic Spider; is it asking you for a common life stone or a diamond? Who knows?
Yokuba (Fassad in the fan translation) becomes this in Mother 3 after falling off Thunder Tower. He can only communicate by speaking through two trumpets, and requires a robot to translate for him.
All of the characters in the original Star Fox spoke unintelligible gibberish, which was helpfully translated into subtitles. Apparently Slippy had time to text Fox while he was getting his aft blown off.
Everyone in the prehistoric chapter of Live A Live. Spoken language hasn't been invented yet.
The Interns from Oddworld: Munch's Oddysee, which have their mouths sewn shut and only speak in mumbles.
Starfy from the Starfy series. He only speaks in squeals, but the other characters can understand him just fine.
Honda Tadakatsu in Sengoku Basara is a Humongous Mecha. So when he talks, it's like his engine is making up sounds, which is usually translated to ".................!!". Other characters (usually Ieyasu) translate what he said.
Mars People: Beep Boop Bop Blap! Red Arremer: Gyah-Gyah! Mars People: Beep Boop Bap! Red Arremer: Gyah!
Ivan from Jagged Alliance series is this for most players, as he speaks in Russian. It is possible to turn on the subtitles, but contrary to all of the other mercenaries (who get English subtitles), his are also in Russian.
Some of his later appearances (Such as in Tales of Xillia) have him communicate with various electrical sounds and/or onomatopoeia instead.
In Tales of Vesperia, most of the Entelexeia can speak the human language, but Ba'ul and Gusios cannot, requiring either Judith or another Entelexeia to translate for them. We also never hear Krones speak, though in his case it's unclear if he can't or merely never does.
In Star Ocean: The Second Story, Ashton's dragons/additional headsGyororo/Creepy and Ururun/Weepy communicate with "grar", "rar", and other similar noises that only he can properly understand, presumably due to having a mental link. In the voiced PSP remake, they say the noises written in their dialogue rather then producing actual roars or growls, which makes their conversations quite comical and able to at least express their emotions effectively. When they have something particularly important to say, however, they take direct control of Ashton and have him say it for them.
One of the voice options in Saints Row: The Third is a "Zombie" voice option, in which the player speaks exclusively in gibberish that's reminiscent of the Tasmanian Devil.
It's subverted at one of the final missions where you get this line:
Zombie Voice: (after some gibberish) I'm fluent in six languages!
Most of the speech sounds in Chibi-Robo are assumed to simply represent speech rather than be the actual words which you read in subtitles. However, the aliens apply this trope. When you first encounter them, their voices are so faint you can't understand them (and you can't read them, either—the tiny voices are represented by tiny text). You have to solve this problem by buying an upgrade that will let you hear them.