It depends. If you count the first two K9s as two separate companions, then the first and fourth Doctors tie for 9 companions each (not counting characters who were only traveling with the Doctor for one story). Otherwise, it's the First Doctor.
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There have been many assistants within the show's run, and even more yet to come unless you only call the classic assistants "assistants". The first assistants, as you term them, may have been schoolteachers Ian and Barbara who traveled with the first Doctor and his granddaughter Susan. They helped bring out the Doctor's more compassionate side to include more than just his family and had influence on his preference for humans as his favorite alien species. If this answer were to include the most recent companions, it would probably make it a bit dated in reference. However, the Doctor has aided and been aided by many, many allies and will continue to be as long as the show is still made.
Matt Smith has been claimed to be the most popular in the childrens program Doctor Who
The Doctor briefly married Queen Elizabeth I ("The End of Time Part I") which is referenced in "The Shakespeare Code" and "Tooth and Claw." He referenced being married at least one other time ("Blink") and had a granddaughter, Susan, suggesting another marriage. The Series 6 episode "The Doctor's Wife" suggests that the TARDIS is in a way his wife, seeing as how connected they are. The Eleventh Doctor marries River Song in the Series 6 finale "The Wedding of River Song." Also, before we were able to watch the Doctor's adventures on our television, the Doctor married a woman. It is known that the Doctor has family, however, they all most likely perished during the Time War. In real life, the Fourth Doctor, Tom Baker, did marry the actress who played his companion Romana, Lalla Ward. David Tennant, who played the 10th Doctor, married Georgia Moffat, who played the Doctor's "generated anomaly" daughter in the 10th Doctor episode "The Doctor's Daughter". Moffat's father is Peter Davison, who played the 5th Doctor.
If you are referring to the ninth Doctor in Doctor Who, then the answer is Christopher Eccleston.
Colin Baker played the Doctor in 1993; however, the Doctors name is literally always the Doctor.