Kimihiro Watanuki in ×××HOLiC is so Keet that eventually Doumeki stops bothering to tell him to quit being so loud and simply covers his ears when Watanuki inevitably gets overexcited. Or annoyed, which happens more often in Doumeki's vicinity. He steadily starts growing out of it due to Character Development, and then abruptly ceases to be this completely when a certain major character dies.
Tendonman from Anpanman. He's the youngest, most traditionally cute-designed, and loudest member of the Donburiman Trio, and was the first Keet of the series, being introduced in the first episode before the other members of the trio.
Yuushirou's very talkative and hyperactive, and rushes thoughtlessly around the place whenever he wants to try and catch up to Yoruichi (which is often). He is eager to please and highly emotional, becoming tearful when Yoruichi leaves him behind or extremely excited whenever he sees her or feels he's done something to please her.
The Sogyo no Kotowari twins from the Unknown Tales arc are small, cute children who engage in playfully dangerous behaviour such as playing fan games with fans that turn out to be energy-firing weapons. Even when they're trying to kill their Soul Reaper master (Ukitake), they never stop being cute. In fact, they even weaponise their cuteness to disarm Ukitake; they almost take his head off with the fan-game before he realises they really are the spirits of his weapons and starts taking them seriously.
A somewhat more mild version is Alistair Cornwell alias Stear from Candy Candy. He's very cheerful, energetic and a teenaged Absent-Minded Professor, but he can also be serious when needed and ultimately falls victim to a truly tragic death in World War I.
Classi9's Haydn is one. He cannot stay still even when performing or leading musicians.
JJ Adams in FAKE, who spends most of his on-panel time gleefully flinging himself at Dee.
Momiji Sohma from Fruits Basket is the embodiment of this trope, even wearing the girls' uniform with shorts instead of a skirt, and seems to be in love with Tohru. He grows up at the end and stops being as much of a Keet, though he's still rather genki.
Shoyo Hinata from Haikyuu!! never runs out of energy.
Shiro of K has this personality and this look... when he's in Tooru Hieda's body. But he's actually a 6-foot-tall silver-haired German adult, and he still has this personality in that body, which is somehow even cuter.
Micchi of Kamichama Karin takes this to whole new levels. His interaction with the rest of the main cast comes dangerously close to flirting (both with Karin and Kazune). This is even though he's a year older and One Head Taller than either of them.
Mizuki from Kamisama Kiss. He's a little older than most examples here but otherwise fits the trope perfectly.
Yogi from Karneval can be overdramatic and flamboyant one moment, then more serious than his "rival" First Ship's female fighter the next. He's not small and cute, though; just loud and cheery. It annoys the heck outta Ineffectual Loner Gareki when his sunshiny personality is directed at him.
A good portion of the male student body falls under this (and the girls aren't any better). The final event in the previous year's Mahorafest was a campus-wide game of tag. Which they aren't repeating this year because there were too many injuries. They were down to a game of hide-and-seek when an army of robots shows up with clothes-stripping lasers. So the teachers simply armed them with anti-robot magic guns and let them loose.
Negi also qualifies when he's able to get away from his Training from Hell and responsibilities as a teacher. When he acquires a time machine during the festival arc and realizes he has enough time to do everything, he goes right into adorable-kid mode. His first impulse was to use the time machine to go back and see the dinosaurs.
Nabari no Ou: Gau, who is hyperactive, passionate and regularly loud. Yukimi's nickname for him is "Tenpa-kun", which is derived from the slang term "tenparu" (lit. "about to blow a fuse").
Son Goku (no not that Son Goku) from Saiyuki. He's this...OH. SO. MUCH! Son Goku is often described as this, especially by fans who have only watched the anime. In the manga, he slowly grows out of it.
Mikogami from Sekirei is a villainous example, and tends to react to anything exciting happening with squealing and comical expressions of joy.
Mizushiro Hizumi from Spiral is "the Devil"... a very, very cheerful and cute one. When he isn't being murderous and/or unbearably angsty, that is - he'll switch between all three moods in a matter of seconds.
Trigun: Vash the Stampede falls into this mode as a part of his Obfuscating Stupidity, but he also seems to take some comfort in the act, especially while enjoying his donuts, which is a sure trigger for it.
Haru from Tsuritama. Cute, eccentric, and least a third of his dialogue is yelled.
Uta No Princesama heroine Haruka Nanami's singing idol Hayato is this. Or rather being keet is his image for the fans.
Q-vier of Valvrave the Liberator (cv: Yuuki Kaji) - a super-assassin Blood Knight with this personality. He's lived most of his life in a secret assassin-training facility and doesn't seem to realize that it's not normal to kill as frequently as you brush your teeth (to quote a similar character from the same writer). The result is unsettling, to say the least, and very sad.
The concept itself is horribly, horribly deconstructed by Kiyoshi Mitarai, though, since he has the looks of a keetand the voice for itbut all it gets him is to be horribly bullied, which later makes him lose his faith in people and join Sensui in his quest...