10 Doctor Who Characters Who Rejected The Doctor
7. River Song
River Song had several one-off adventures with the Tenth, Eleventh and Twelfth Doctors between 2008 and 2015, but when she was offered the chance to become a permanent companion, she turned the opportunity down.
After losing Amy and Rory to the time-displacing touch of a lone (and rather sneaky) Weeping Angel, the Eleventh Doctor develops a major case of the sad, sad face, and River - noticing this - tells him that he must never travel alone.
And so, the Doctor does the obvious thing, and asks her to tag along with him. River does seem to contemplate the idea for a second or two, but she then politely refuses, telling the Doctor that she will always be there for him... just not 24/7.
"One psychopath per TARDIS" is how she puts it. Fair point.
The Doctor and River's timeline is so wibbly-wobbly complicated that it would probably rip a hole in the universe if they travelled together constantly, so even though the Doctor needed a shoulder to cry on at this time, it was definitely the right decision for River to stick to her infrequent TARDIS adventures, rather than making it an indefinite gig.