This one didn’t take long to pick. There is, of course, Barbara and Ian, the Doctor and Rose, the Doctor and River, Amy and Rory,
But there is one pairing that is above them all. The Doctor and his TARDIS.
This has been brought up in the show a number of times how the TARDIS is connected through a psychic link with the Doctor, and is alive, though I think the meaning in the very beginning meant powered with electricity, but it later was clear that it is a sentient being.
For a long time, the Doctor thought he had stolen (sorry, “borrowed”) the Tardis to escape Gallifrey. It turned out that the Tardis stole him as well.
I love how a lot of this is brought up in the excellent episode, “The Doctor’s Wife” where the 11th Doctor, Amy and Rory end up outside the universe and a being called House takes the “soul” (for the lack of a better word) of the Tardis out and places it inside a woman so he can take over the blue box. And the Doctor and Tardis start talking to each other.
Anyway, The Doctor’s Wife is a great first episode (along with Blink).
Next in the Doctor Who 30 Day Challenge – Favorite New Who Series