The term usually isn't used on screen. More often they used other terms, like "friend" (as in "I'm the Doctor, and this is my young friend Ace"--almost a catch-phrase for the 7th Doctor) or "assistant". And sometimes they write in-jokes for the fans where people seem to go out of their way to avoid the word.
I don't know if anyone has made an official count, so you'd probably have to download all the transcripts from one of the sites that collects them and grep them all for a definitive answer.
But here are some examples:
- In the 7th Doctor story "Battlefield", the Brig says, "He has many faces. And many companions. This must be the latest one."
- In the 10th Doctor story "Army of Ghosts", Yvonne Hartman in "Army of Ghosts" refers to "the Doctor and his companion" as the "pattern".
- In the 10th Doctor story "The Stolen Earth", Harriet Jones calls Martha a "former companion."
- In the 10th Doctor story "The Next Doctor", the Doctor asks Jackson Lake (the "next doctor") if he could be a companion (although I think that was in a deleted scene).
also "The Next Doctor" features the word several times: Jackson: "This is Rosita, my faithful companion." Jackson: "the Doctor's companion does as the Doctor says. Now off you go." the Doctor: "I'm your companion." Jackson: "All those bright and shinning companions."
- In the 11th Doctor story "The Pandorica Opens" the Supreme Dalek calls Amy "your companion"