What are your Doctor Who head canons?

Merlin the Wise
This incarnation of the Doctor may have taken a bibliothecarial attitude to his past lives and inveigled himself amongst the High Evolutionaries within the Matrix with the intention of monitoring and assisting his past selves wherever he could as part of his custodial responsibility as one of the more elderly (and therefore wiser) iterations of his life. His interactions with his past incarnations and mindful tampering may have lead to his expulsion from the microuniversal complot and further involvement with the alternate dimensions existing both in parallel and adjunctly to the third universe, leading to his fascination with the sorcerous continuum in which Arthur and his knights dwelt. It is suspected that the events depicted in various Marvel graphic stories are distorted accounts of various battles between Arthur and Morgaine spanning over a relative twelve centuries and by this time the Doctor had acquired an ability unique to this incarnation that allowed him to transphase between various renderings of himself and interact through these facets with the inhabitants of the other dimensions. It is unclear where Merlin may have acquired this skill set, but given his affinity with the Matrix and Arthur's universe it's not entirely inappropriate to suggest that in his aged wisdom he has adopted several magical powers as a means to facilitate his ends in much the same manner as his fourth self's sonic screwdriver or his seventh self's umbrella. He much like his successor would spend a great deal of time with Arthur and his knights, experiencing most (if not all) of his adventures in reverse order before regenerating. He would seem to have abandoned the normal universe to its own devices prior to his estrangement (presumably to escape Morgaine's trap in the ice forests of Brecelainde) to the surface of Antykhon with a new face.

Muldwych
An incarnation with a head of unruly red hair wearing a tatty embroidered Afghan coat and a floppy hat of brown felt with a saffron bandana who forced himself free of Morgaine's trap on Breceliande using the astral plane. The effort subsequently marooned him on Antýkhon in N-Space. His decision to send the moribund King Arthur aboard the King's Hall ship would earlier leave him responsible for the events of Battlefield as faced by his seventh incarnation.

"Audio Visuals" Doctor
A gentleman, a scientist and an explorer with an insatiable sense of curiosity, this iteration was known for always keeping his moral imperative even in the face of the abject and seemingly ubiquitous evil prevalent throughout his travels across the universe. He was an ardent devotee of tea and was known to be impatient with misguided or purely evil individuals; often irritated at their lack of prudence and Weltanschauung, rather than out of a more tralatitious sense of righteous moral indignation.

Dr. Who
He had been defeated and become trapped in one of the Toymaker's celestial games as a plaything, forced to live out a portion of his incarnation as a doddering English artificer from Earth in Tardis as seen in AARU's Dr. Who and the Daleks and Dalek Invasion Earth 2150 A.D. It's conceivable that he and his companions eventually became aware of their predicament and engineered their escape from the Celestial Toyroom using the real TARDIS. This incarnation of the Doctor was known to have travelled with seven different companions including, but not necessarily limited to several exaggerated caricatures of his previous companions from the fantastical realms of the Celestial Toyshop to his good colleagues Preponderant Jormunaire Kharis (Christopher Lee) and Archtreasuerer Wreave Slyne (Vincent Price) of 14460s Irulaius to whom he maintained a tender confidence.

"Richard E. Grant" Doctor
Self-indulgent, manipulative and arrogant to a fault, this incarnation is nevertheless a provocative figure who serves as a powerful force for inspiration and change across much of the universe. His predecessor met a rather gruesome end when he was shot in the back upon arrival at the London docks in 1949 by a woman named Mestizer during the Cabinet of Light Affair. He spent several months lying low while Honoré Lechasseur conducts his investigations for him in the interregnum before departing to future adventures.