10. Lynda (With-A-Y)
It's not always the Doctor turning potential companions down. Lynda (with-a-y, as she tended to introduce herself) was introduced in the Series 1 finale Bad Wolf/Parting of the Ways, meeting the Doctor in a deadly version of Big Brother. The Ninth Doctor takes a shine to her even before they escape into the repurposed Satellite 5. She's also courageous, staying behind and helping the Doctor against the oncoming Dalek horde by reporting on their movements. Such a trait would have been really useful in a companion. Furthermore, being such a key part of the Doctor's plans to hold off the Daleks shows how quickly she gained his trust, even when he was critical of the rest of her society. However, she never got the chance to tag along in the TARDIS. Lynda's willingness to help the Doctor ended in her death, as the Daleks tracked her signal and exterminated her during their siege on Satellite 5. So close but yet so far.