In general The Doctor prefers the company of us humans over pretty much anyone else. Nearly every traveling companion of his has been human and despite the vastness of the universe most of them are also from Earth. Today we celebrate a few of the rare non-human companions that have joined him over his many lives.
Kamelion The shape-shifting android Kamelion was a very brief companion of the Fifth Doctor, appearing in just two episodes. This was mostly a technical concern. Doctor Who showrunner John Nathan-Turner had found the robot somewhere in a prop department, and the robot was cumbersome and hard to manage. On top of that Kamelion's software designer, Mike Power, died in a boating accident shortly after the robot was accepted to appear, which further limited its use as Power left behind fragmented operating notes. Peter Davison hated working with the prop and was glad to be shut of it. Kamelion has made a few appearances in other media, including "fathering" a child with the Tardis itself in a Sixth Doctor short story, "The Reproductive Cycle".
The Master I'm intentionally leaving out both fellow Time Lords like Romana and characters like Leela who are humanoid but not descended from Earth. According to "Zagreus" the Time Lords are partly responsible for the creation of all humanoid life in the universe anyway, or to quote the Eleventh Doctor, "You look Time Lord". As such I'm focusing on companions who are more obviously not human.
The Master is a special case, though (isn't she always?). In the webcast Scream of the Shalka that was originally to be a revived series until plans were finalized for the current show, The Master is a full-time traveling companion to The Doctor, and is for some reason trapped in a robot body that can't leave the Tardis. Voiced by Derek Jacobi (who would return to the role in "Utopia"), this Master is a somewhat reluctant but nonetheless loyal member of the crew that is instrumental in defeating a race of psychic vampires. The Shalka universe has been cosigned to a distant alternate universe, but it remains a glimpse of something altogether amazing that could have / maybe happen.
Compassion One of the hardest companions to categorize is Compassion. She was originally a human from a distant colony that through a series of bizarre accidents became a living, sentient Tardis in human form. At the time the Eight Doctor had lost his own Tardis and the Time Lords were looking to capture Compassion to breed their own fleet of sentient ships. This led The Doctor, Fitz and Compassion to go on the run using her as a portal through space and time. Eventually she returned The Doctor to his own ship and left to explore the universe. Like the rest of the Eighth Doctor novels this likely happened in an alternative timeline.
Handles Though the head of the Cyberman affectionately referred to as Handles only has one on-screen adventure with The Doctor it was a truly memorable one. The Doctor used the head as a kind of super smart phone / assistant, but it's very clear from the way he speaks with Handles that he's fond of the Cyberman. Even more so, Handles himself actually displays emotions near the end of his long life with The Doctor, including a grudging thanks for some additional comfort. It's implied that Handles has actually been with The Doctor longer than any other companion.
Majenta Pryce The Tenth Doctor took aboard Majenta in the pages of Doctor Who Magazine in comic form. She was a green-skinned Vessican who was never all that trustworthy, but she did save The Doctor's life on numerous occasions even if it was largely from the consequences of her own dirty dealings. Her path was one of the most heartbreaking of any companion, involving memory loss, body conversion and eventually her death to undue her own mistakes. Nonetheless, The Doctor made sure she was not forgotten.
C'Rizz A Eutermesan from a divergent universe where linear time did not exist, C'Rizz was reptilian with the ability to change his skin tone. He was broody and prone to melancholy because he was forced to kill his fiancé after she turned into a giant bug, but was happy enough to join the Eighth Doctor and Charley as they moved from planet to planet seeking the Tardis and an escape. C'Rizz eventually traveled to our universe with them, where he often had a difficult time fitting in. He was easily manipulated, but fiercely loyal and protective, particularly of Charley.
Frobisher Another shapeshifter, Frobisher is a Whifferdill from Xenon and is one of the most prolific of the comic companions having traveled with both the Sixth and Seventh Doctors, meeting the Eighth and being vital to saving the lives of his fellow companions in the Prisoners of Time epic story. Frobisher started life as a con man who decided to turn into a detective, and preferred to spend most of his time in the form of a talking penguin. For a brief while a disease confined him to the form but he eventually regained his full abilities. More even than most companions he had a habit of getting himself into ridiculous situations, even being named a god once, but when the chips were down he was often one of the most dependable companions.
Destrii The Eighth Doctor alone has had a recurring non-human companion in every medium he has appeared regularly in. His final companion was Destrii, an Oblivioner from a race where individuals were genetically modified so that no two were alike. She started out as an enemy, switching minds with The Doctor's companion Izzy and then being involved in a long quest to restore the two. She eventually turned on her own people to save Native American children in 1875, suffering a severe beating and earning The Doctor's trust. She traveled with him for a while as his muscle, being a keen fighter. It was originally planned that she would witness the regeneration of the Eighth Doctor into the Ninth and travel with Nine before Rose for a series of adventures. The BBC vetoed the idea and the Doctor Who Magazine stories of the Eighth Doctor end with the two friends walking off into the distance with Destrii's fate ultimately unknown. She is not mentioned among the companions the Eighth Doctor toasts before he regenerates.
Madame Vastra The leader of the Paternoster Gang is the Silurian Madame Vastra. Though she's backed with Strax and Jenny, it's Vastra that serves as The Doctor's main lieutenant in large encounters. In many ways she is the revived series' version of The Brigadier, a solid presence that can be counted on for anything. It's Vastra who protects and aids the Twelfth Doctor after his predecessor regenerated, much as the Brigadier did for the Fourth, and she was the first person he recruited when he went to war on Demon's Run. There are few companions who have proved themselves more capable, and only one non-human one...
K-9 It couldn't be anyone else. The robot dog K-9 traveled mostly with the Fourth Doctor but has been at the side of numerous other Doctors on occasion as well as serving as an assistant to both President Romana and Sarah Jane Smith depending on the model. He is also the only non-human companion to receive his own spin-off series in Australia. Aside from The Doctor, his Tardis and the sonic screwdriver K-9 is probably the most recognizable aspect of Doctor Who, making many appearances in merchandizing and appearing as a parody in both Fallout 2 and South Park. He's in a class all by himself, even in the niche world of non-human companions.