For a long time most of the progress in CGI was towards realism. Ever since the form evolved from mathematically reconfigurable 2D shapes to illusory 3D forms with ever more naturalistic lighting, there has been a tendency to use naturalism as a standard, due to technological limitations, a double purpose as special effects and a general stylistic conservatism. CGI started as basic and sometimes cartoony to become more naturalistic as time goes by, but steering clear of the uncanny valley when possible, only to reach a point where the filmmakers had to push the boundaries in order to look 'newer' than previous films. I'm going to start with a couple of notable short examples from before Toy Story and some features that progressed towards cartooniness after.
The first movie to break the mold stylistically was Jimmy Neutron: Boy Genius in 2001. This movie doesn't contain particularly cartoony animation but the art design is rather unusual. The character's proportions are quite far removed from reality and Carl has black pupirises with his glasses outlining his otherwise nonexistent eyes. The characters have very large heads and thin necks. It doesn't feel like a golden age Looney Tune or something from UPA (Carl's eyes aside)but it feels like a 90's kid's show translated to CGI.
I won't say much about the Veggie Tales movies because Veggie Tales started as a TV series before Toy Story came out and I'll be talking about it in another post.
The next movie to change things was a much more representational and much more famous movie, Ice Age. This movie arrived at such a time that it managed to innovate towards both realism and cartooniness at the same time. Realism can be seen in the textures and fur with multiple textures and the lighting that's more sophisticated than before. Cartooniness can be seen in Scrat's movements somewhat but especially in his bulging eyes and of course, the jagged ice that acts almost like a character itself.
2003 didn't have much cartooniness but 2004 did. In fact, three films contributed to less realistic CGI in slightly different ways.
Appleseed isn't particularly cartoony in forms or animation but it is the first cel shaded mainstream animated movie and deserves to be noted if only in that regard.
Next is the Dreamworks film Shark Tale. This film is slightly more stylized in setting than its Pixar predecessor Finding Nemo. It is most notable for its characters that look like they're from an awkward Looney Tune parody with scarily human-like faces on sea creatures.
Next we have a film that's from Denmark, based off of a radio show, Terkel In Trouble. This movie didn't have much influence on mainstream animation in other places but it set the stage for later Danish CG. The characters are simple with Simpsons-like bulging eyes, arms and legs that default to curves and unusually large mouths even for cartoon characters. This is the most cartoony CGI film made at the time and for fans of adult black humor they should find it at least decently fun. The film, while notably cartoony is rather conservative in animation and in the forms of the characters themselves. This would develop more later on.
You likely already guessed the next one. The Incredibles is notable for being Pixar's largest deviation from realism in terms of human character design thus far. The idea was to keep the humans from looking too realistic so they were designed a bit more sculpturally than Pixar's previous human characters. This movie is different in that it was the first CG animated film to combine less realistic character forms with with normal hair.
2005 saw three films with notable cartoony characteristics. Doogal was influenced by stop motion but the film doesn't seem to do anything besides that so I won't include it.
The first of the three films is Robots. This film is the one that really takes an unrealistic art style and runs with it. While the animation itself is mostly rather non-cartoony aside from some of the mouth movements that seem too implausible for such mechanistic looking robots, the character designs certainly are and this film contains sequences that would inspire many of the later films of cartooniness, the chase scenes in particular. And who could forget the exotic transportation via a large Rube Goldberg apparatus?
Then we have Madagascar which takes a designier approach to animals than any animated film that came before. It has a lion that spends more time standing upright than on all fours. All of the four main characters are more artistically designed than any CG animals in a movie before with pushed designy shapes that call attention to themselves and generally toned down emphasis on fur. This movie is also different from an animation perspective with lots of cartoony poses and things don't move as realistically or predictably all the time as in films from years past.
Lastly of the year we have Chicken Little. It has somewhat cartoony animation again so I'm including it. After this that won't be enough to consider something on the cutting edge so I'll focus on the stuff that changes.
Next year, in 2006, there aren't as many changes design wise to cartoony CG animation. The real change is in the content which becomes for once more surreal. In Cars we see something that we hadn't seen in CG features yet, talking cars. Then in Everyone's Hero there is a talking baseball and baseball bat. One film stands out though, from a visual perspective.
Flushed Away attempts to create a claymation-like style in 3D CG. It didn't work as well as they'd hoped because I thought it was just a less realistic CG style, not an attempt to look like claymation. In changing from representing features faithfully to doing a less realistic approach like with sausage-like lips and ears that are impractical basic shapes it nonetheless breaks new ground.
2007 is a year that has relatively little to offer as far as cartoony CGI. There's independent films which are totally or partially CGI like Flatland the Film or We Are the Strange but these are more experimental than cartoony. They don't have caricature in them and cel shading had been done in years prior so there's little point in mentioning them. Meet the Robinsons has futuristic designs that are cartoony and there's some artisticness to the character's proportions but they all have normal features as far as CGI goes.
Moving onto 2008 we have what could be considered the first truly cartoony CGI feature in both art design and animation. The film is Horton Hears a Who, an adaptation of the Dr. Seuss picture book. It has essentially set the stage for most cartoony CGI that came after with the cartoony designs and flexible rubbery motion but characters having every strand of fur and hair.
Horton Hears a Who isn't the only notable film from 2008. There is the Norwegian film Kurt Turns Evil. It has an interesting sort of modern art character design and design in general but the movements are organic and thus don't mesh with the style and complete the cartoony feel.
Next we have Journey to Saturn, a Danish cartoon with characters who have round heads, round simple ears, cartoony round projecting noses and other traits that were absent from Terkel in Trouble. This film has the first flat eyes, this time with small irises.
After this we move along to 2009 where we see an important entry in cartoony design from a studio that would embrace the idea later on.
Sony Pictures Animation made Cloudy With a Chance of Meatballs, a film that bears little resemblance to the book that it's based on aside from a similar food weather premise, and changed the game again. The characters have a more UPA and other 50's cartoon influenced aesthetic. The kids are proportioned similarly to Gerald McBoingBoing proportions and the lipless mouths, when showing enthusiastic glee open up with connected Us on both the top and bottom. The cartooniest part of the design though is definitely the hands. These hands are not even close to being realistic. Some characters have long fingers like Flint Lockwood and the police chief has hands with no clear wrists connected to his long Popeye-like forearms. The eyes are flatter than in other CG films just like in JtS which is a trend that would show up elsewhere soon.
2010 had a couple of somewhat cartoony CG films. There was Despicable Me and Gaturro.
Despicable Me has adult people with stick limbs and exaggerated craniums but the children are rather generic CGI characters. The minions, aside from being incredibly annoying, are yellow humanoids. They didn't take them to very cartoony lengths though.
Gaturro is based off of a comic strip and in spite of the cartoony designs inherited from the comic strip which probably are the most extreme up to that point, the animation isn't generally cartoony from what I can tell. There is the main character's eye spinning around but screwing around with eyeballs was already done in Cloudy.
2011 has a couple cartoony CG movies.
Kikoriki is based off of a television series but is notable for its sphere based chararacters. Characters based on spheres goes back all the way to early 20th century comics.
Freddy Frogface is a movie with a sort of deliberately ugly storybook-like caricature. The film has head shapes that focus more on the overall shape of the head than traditional proportions so it's breaking new ground in that respect. It turns out that this movie came out slightly before Gummi T, known as Ivan The Invincible in the US.
Ronal The Barbarian continues the Danish tradition. It has a rather cartoony skin rendering and the faces have interesting features like noses and the rounded ears. Aside from the heads, the bodies aren't all that cartoony in terms of features. But what this movie lacks in a cartoony look it makes up for with cartoony soul.
Daddy I'm a Zombie has characters with ultra-thin bodies and limbs but is otherwise rather non-cartoony.
2012
The year 2012 brings about an iconic series of cartoony CGI.
Hotel Transylvania is one of the cartooniest films ever made up to that point. It has full body caricature, poses and movements. The reason this film is so cartoony is that it was helmed by Genndy Tartakovsky who was the creator of Dexter's Lab and Samurai Jack. You'll see the sequels to this film listed here where the action becomes even more cartoony.
Tad The Lost Explorer is a step forward for Spanish CG but it's ultimately less cartoony then the norm for its time so I won't show embed a video for it.
2013
The first major cartoon CG movie is Otto The Rhino. It's like Harold and The Purple Crayon in the style of Freddy Frogface. It doesn't really do anything new but it's worth mentioning.
There is a Chinese animated movie that is probably the most cartoony cel shaded CGI done up to that point. I highlight this and not The Painting because the other is more artistic than cartoony.
https://movie.douban.com/trailer/162470/#content
There is the sequel, Cloudy With a Chance of Meatballs 2. It creates a whole ecosystem of talking food so I'll include a trailer for that reason alone.
Minuscule: Valley of the Lost Ants is a film with animated insects done over live action settings. The characters are insects with exaggerated cartoony dot eyes and interesting but simple personalities. It's based on a TV series but it is the most important cartoony CG movie released this year due to its innovation relative to other movies so I'm including it.
2014
I won't say much about The Lego Movie since it's more toyetic than cartoony.
Stand by Me Doraemon is a movie based off of the Doraemon comic and television series. The characters are of the rounded comedic manga type with exaggerated expressions.
Mr. Peabody and Sherman is based on the old TV show of the same name. It's more of a polished version but it has some slight cartooniness. The only reasons I'm including it are Peabody's still cartoony design and the gags.
The Book of Life
This movie is a dimensional version of Jorge Gutierrrez's style. His 2D style is very flat and stylized, so seeing his work in 3D was interesting. All the characters have some level of stylization to them from the people at the museum to the main characters of the story. Hairdos are exaggerated, skulls have stylized nasal passages, and there is a lot of the big around the jaw heads which you'll see later in every yeti in Smallfoot. I don't know where that trend began, but it has been around since The Flintstones.
Asterix and the Mansions of the Gods This movie is a CGI adaptation of the Asterix comics series which looks pretty much like the original but in 3D CG. The rendering is the most notable part, just right for the exaggeration they're going for.
Mortadelo and Filemon: Mission Implausible
This movie is one of the more advanced cartoony CGI films. Like Asterix and Gaturro, it's based on a comic, in this case a Spanish comic strip. However, this film has more detailed anatomy to go along with the cartoony facial features which are more anatomical than before.
2015
Hotel Transylvania 2
This film is similar to the first Hotel Transylvania movie but it has more ambitious large scale gags.
The Peanuts Movie
The Peanuts Movie is a new sort of take on CG comics adaptations. There's the dot eyes, previously only seen on Carl from Jimmy Neutron but this time they're on every character. The mouth expressions are drawn on the outside and modeled on the inside. The animation is done on lower than usual frame rates for an old fashioned 2D feel and the rendering for hair and clothes is slightly toned down though still within CG fashion.
The Good Dinosaur
I haven't seen the movie, like many others on the list, but it has an especially short child for his age and the dinosaurs are unusually cartoony for the environment.
Snowtime! Some might consider this more of a model-like approach with the painterly looking textures and the motion of the characters that doesn't make much of a statement. It's notable mostly for the character's pupirises which take up more space than the whites of their eyes.
2016
Bad Cat This movie is a hard hitting animation from Turkey with moderately cartoony designs. The teeth of the main cat are the biggest exaggeration in it. The movie is packed with sadistic gags one after another.
Sausage Party This film is notable mostly for its use of anthropomorphic food with classic dot eyes and cartoony three fingered hands. The human designs have slightly pushed proportions, more aesthetically pleasing than the Mort and Phil ones but don't push the cartoony extremes.
The Angry Birds Movie This movie is based on a mobile game that's already somewhat cartoony though the game is 2D and the movie is a souped up CGI version of that. The designs are sort of catoony but less so with the realistic feathers. The real cartooniness is in the poses during the fight scene in the film's third act when the characters storm the green pig's castle.
Storks This film has parts that are more cartoony than others The main characters are relatively normal, though the stork has some wacky poses. The pigeon character and the wolves are the cartooniest part of the movie. The pigeon has some very animated poses and the wolves are exaggerated looking and moving creatures which can form into large objects of their choosing.
2017
Captain Underpants This film takes the illustrations of the children's book and renders them faithfully in CGI which makes it very cartoony indeed. This is one of the more unique styles in these cartoony films due to the character designs which are more tubular than the others which are more traditional using spherical or cartoony skulls and hands usually have more pronounced palms. The general design of the movie is unrealistic but only the proportions are really that cartoony.
Ferdinand This movie is notable mostly for the poses and the smooth fur which is a break from the realist tradition. I haven't seen the movie, like many of the others so there's not much else I can say about it.
Hotel Transylvania 3 This second sequel to HT included more exaggerated elements like the super tall cruise ship and the island where the characters end up. The characters have even more pushed poses than in the last two films.
Smallfoot This film has yetis which look similar to some characters from The Book of Life but with very individual fur patterns that are very naturalistic. The human characters are pretty stylized but some in a less cartoony fashion and they are unusually small to make the contrast between the humans and yetis more dynamic.