It is slightly difficult to compile a list of 'the best dogs in video games', since dogs are basically the greatest animals to ever exist in the history of mankind and every dog is the best dog.
However, just in case you are also someone that would play a game just because a dog plays a major role in it, here is a list.
The pomeranian is the first non-DLC predator available by default at the beginning of the game.
It is very small, and as a result has rather low base stats, which makes it difficult to use. However, this also means it requires fewer calories to survive and 'level up', developing into a formidable predator.
The Mabari Hound sidequest, upon completion, earns you the undying loyalty of this pureblood Mabari hound whose default name is 'Dog'.
Not only is Dog a powerful ally in battle, he also loves you unconditionally. One of the mechanics of Dragon Age: Origins is an 'approval' system that will cause you to either gain or lose allies. In the case of Dog, it doesn't matter what you do, he will always stay by your side.
Shadow is a Malamute dog, partner to Dead to Rights: Retribution protagonist Jack Slate.
When Shadow was just a puppy, he was rescued by Jack's father, Frank. Shadow was put through the K-9 program. He was considered to be too vicious, however, and was failed out. Frank instead brought Shadow to the 87th Precinct of Grant City, where Shadow helped Frank and Jack solve cases.
After Frank is killed, Shadow fights alongside Jack. Shadow can sneak around and can do silent kills from behind, ripping out throats. As he is basically made of pure muscle and ferocious loyalty, he can also charge in and bowl over enemies.
Riley is a German Sheperd military working dog and part of the US Special Operations unit that is featured in the game, the Ghosts squadron. He is based on a real Navy SEAL trained K9 named Ruger (motion-capture).
Riley can sniff out explosives, and assists the player in combat. Riley can even take down low-flying enemy helicopters by jumping into them and getting at the pilot.
According to the developers, Riley was added because the developers wanted to assure that the players felt emotional attachment to the central cast. This definitely worked; Riley was a huge hit with the fans.
Hewie is a White Shepherd that protagonist Fiona Belli saves from captivity. In gratitude, he becomes her steadfast ally, aiding her in her attempts to escape Belli Castle.
Like a real dog though, Hewie should be both well trained and taken care of in order to maintain a good relationship between him and Fiona. Fiona has to praise or scold Hewie when he obeys or doesn't obey commands. He also should be praised and/or healed when he is injured. This increases Hewie's friendship level with Fiona. Repeatedly mistreating Hewie will result in him ignoring or outright attacking her. The level of their friendship is very important as it will affect which game ending the player gets.
You meet the dog at the very beginning of the game during the protagonist's childhood. You (the protagonist) and your sister Rose rescue the dog from a bully. The dog follows you home and becomes a part of your family, staying by your side through thick and thin.
Game mechanics wise, the dog helps you locate chests, quest objectives, etc. Most importantly, the dog is directly influenced by your moral choices as the player. Over the course of the game, as the protagonist's alignment shifts to either 'good' or 'evil', the dog's appearance will change to reflect that.
A good deal of your dog's life will be deeply affected by your decisions in-game, further down the line.
Dogmeat is an Australian Cattle dog, a potential companion and ally that can be found in the Scrapyard. He is an homage to the Dogmeat from the first Fallout game.
Dogmeat is unwaveringly devoted and is a powerful asset in a fight. However, while he is very resilient, he can be killed. This has motivated players to take advantage of the 'Puppies!' perk that enables Dogmeat to be reincarnated as a puppy upon death.
Chris Avellone, Chief Creative Officer of Obsidian Entertainment and creator of the Fallout Bible said:
For many reasons, Dogmeat is arguably the most successful NPC companion ever.
Koromaru is an albino Shiba Inu and is modeled after the famous Hachiko, the loyal dog who waited for his owner to return, even years after his owner's death. Koromaru is a rare example of a non-human that has the strength of spirit and ability to summon a Persona (Cerberus). As a result, Koromaru joins your party in its fight against the Shadows and other apocalyptic enemy forces.
In Persona 3 Portable, you are able to establish a proper Social Link bond with Koromaru which really shows you how loyal he is. As you strengthen your bond with him, he comes to really trust and care about you and you will get to see the extent of how gentle and wonderful this dog is.
The 'social simulator' aspect of Persona results in the best kind of payoff with your interactions with Koromaru since, as you interact with him and gain his affection, he will bring you things and snuggle you.
Technically, Amaterasu is a wolf. However, in Japan dogs and wolves are pretty much interchangeable. In some areas of Japan, wolves are referred to as 'yama-inu' which literally translates to 'mountain dog'. (This accounts for why every character in the game thinks Amaterasu is just a white dog).
Amaterasu is, in Shinto myth, both goddess of the sun and universe. In Okami she is the protagonist, incarnated as a white wolf. (This taps into a conflation of meaning here: 'okami' can mean 'wolf' and also can translate to 'great god').
Amaterasu, nicknamed 'Ammy' by Ishaku and Isshun, is a friend to all living creatures. All throughout the game she tries to help every living creature she can. One major game mechanic is a Praise system: every time she earns the belief of those that she helps, through completing quests, sidequests, and by feeding animals she meets, she becomes stronger.
While she is technically not a canonical character, Mira deserves a special mention since she is both a Shiba Inu and an evil mastermind.
Mira has appeared only in joke endings. Her debut was in the 'Dog Ending' of Silent Hill 2. (It is unlocked with the Dog Key, found in a dog house beside Jacks Inn). She also appears in the 'UFO Ending' of Silent Hill: Origins and Silent Hill: Shattered Memories, as well as in Silent Hill: The Escape and Silent Hill: Book of Memories.
In the case of Silent Hill 2's Dog Ending, it reveals that all the events of the game were the work of Mira, a dog sitting in a control room, pulling various levers. The protagonist, James Sunderland, falls to his knees in shock at this revelation. Mira simply walks over and licks his face.
Mira has been credited with playing the guitar of Waiting For You alongside Yamaoka Akira. She was also credited for making the screenplay of the Silent Hill: Downpour trailer. Obviously she is very talented and is recognized by the developers as one of the most important characters in the Silent Hill series.