To tell the truth, this is more fodder for my noncanon Who thread than the pet theories thread, but this is a more community thread sooo...
You know how they do "The x Doctors" stories and one of the problems is always that the actors have gotten older--sometimes to the point of being dead? It got me to thinking about "Quantum Leap" and how they could borrow that premise for a Who story. There's some kind of TARDIS malfunction and the Doctor finds himself slipping through past incarnations. Have the current actor play him, but stick him in past costumes and have the occasional reflection, a la young John Hurt. You could even have most of them be "lost" stories. 1, 3, 4, and pretty much every Doctor from 6 on have spots where they were without a companion. And they used 2 in "The 2 Doctors" so one could argue the Time Lords used him for a couple stories between "The War Games" and "Spearhead From Space." I'm trying to remember "The 5 Doctors." Wasn't 5 separated from his companions for that story? Even he could have had an adventure before going back to the TARDIS. I don't know if you just steal the "Quantum Leap" idea or do a full-on homage and have the companion stranded in the TARDIS, using the TARDIS' telepathic circuit to give the Doctor assistance like Al.
It could be fun.