Be warned, many a word lie beyond ye point.
I'm a bit believer in the idea of Looms, at least until the end of
Lungbarrow when the curse of sterility is lifted. After that point, it's pretty fair game.
The Merlin incarnation that's mentioned in
Battlefield and later appears in the New Adventures seems to have two completely different descriptions, depending on whether you read the comics or the novels. I like to reconcile them by thinking that the DWM!Merlin is one of the Doctor's oldest incarnations who has taken a rather bibliothecarial attitude to his past lives. I like to think that he inveigled himself amongst the High Evolutionaries within the Matrix as a minder of the cosmos, someone who still travels and experiences the wonders of all the galaxies, but still has obligations and duties to perform back home. Judging from his sudden disappearance prior to
The Final Chapter, he was probably expelled from his position or chose to leave to the sorcerous continuum where Arthur and his knights lived. He lived many of his adventures in reverse as befitting Arthurian myth (insert in-joke about William Russell in
The Adventures of Sir Lancelot here) and by this point has probably abandoned the normal universe to its own devices. His ultimate fate is unknown, but he will eventually regenerate into the Muldwych incarnation (probably in the company of the Knights since Ancelyn knows that his face changes) who will carry on in the role until he is trapped beneath the ice forests of Brecelainde.
Following a similar vein, the Richard E. Grant incarnation for me is an iteration that occurs in the Doctor's own future. The reference to Andy Warhol "wanting to paint all nine of [him]" can be interpreted as a visit made by the Eccleston Doctor during his long period alone. Identically, the Ninth Doctor's reference to his ears while implying a recent regeneration could easily be a reference to either a niggling physiognomic weakness or (as someone on the forums pointed out once) something as simple as a haircut. I always thought it'd be an interesting twist if the REG Doctor's employers turned out to be the Daleks, instead of the Time Lords. Using him to maintain the temporal stability of the universe, while they dealt with a catastrophic crisis back on Skaro.
The various continuity gaps where companions disappear or should appear but don't, I explain away through fanon. For example, the Sixth Doctor, Peri and Frobisher had a number of adventures before
The Trial of a Time Lord including a trip to the Khyber Pass and an encounter with Gunga Din, the Boston Uprising, encountering a Silurian shelter that has been coated in conductive gold in sixteenth century South America and a meeting with the Japanese Robin Hood, Ishikawa Goemon. Another example is the First Doctor's period alone, which I think was granted just after
The Three Paths where he's given the opportunity to return to Gallifrey yet refutes it, electing to continue his travels despite the great cost it may have. It leads rather splendidly into the events of
The Tenth Planet, but I like to think that (given his reference to Ben Jackson in the bookends for
The Witch Hunters) that he was abducted at this point for
The Five Doctors and was granted exclusive privileges by Rassilon that allowed him to attend to several unresolved threads before his death. I'd like to think that the process was technologically analogous to a spell, which slowly began to fade over time until the First Doctor eventually reassumed his place in history.
The Peter Cushing incarnation 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 the films. 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, which was hidden away from him. I got a bit carried away with the worldbuilding when I found out this theory existed, so in my mind 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.
The same is true of Nick Briggs's
Audio Visuals Doctor, who I think is too good not to be part of canon; he's a future incarnation we haven't seen yet. There are many like him, ranging from the Nick Scovell incarnation's lone studies to Richard Griffith's Doctor traipsing around with Kate Tollinger in 1992.
I love the idea posited on "Rassilon, Omega and that Other Guy" that when the Doctor's penultimate incarnation regenerated, he underwent mitosis and split into two separate entities which became his final incarnation and the Valeyard respectively. Judging from what we learn in
Trial of the Valeyard, it's possible that this was a result of the Doctor wresting to grips with his own mortality by experimenting upon himself in an effort to prolong his lives. It would work after a fashion, the Final Doctor would live for longer than all of his previous lives combined, but he would wreak this Jungian shadow's existence upon the universe as a consequence. The Valeyard was eventually found on Eta Rho, had his encounters in the Shadow House with who I suspect was the Grandfather Paradox before he was transferred to Shada. He was able to gain access to Time Station Zenobia by preying on the High Council's fears for the future, posing as the tragic survivor of a doomed future, the High Council requested current Matrix prognostications in order to cross-reference this new information and discovered several possible futures in which the Time Lords would be obliterated. Ensnared by his duplicitous masquerade, the Valeyard proposed converting the Celestial Intervention Agency's superorbital time station Zenobia into a judicial installation capable of apprehending, judging and sentencing the temporal apostates supposedly responsible for these future incursions -- and this would eventually lead to the arrest of the Sixth Doctor. It's an event that sounds as though it wasn't initially meant to happen, but became part of the canonical timeline once it had.
As to who this final incarnation is, I like to think he's the self-appraised "grandfather of the universe", portrayed by Michael Jayston and travelled around with companions named after the original production team. On the other hand, the Gods of the Fourth and the Kingmaker seen in
Death Comes to Time have such a radically different (and smaller) take on Gallifreyan society that I can't help, but feel that maybe the Final Doctor lives in a universe very similar to the one we see there. The Fraction, the Kingmaker, Tannis, the Minister of Chance, Nessican and all.
There's a lot there, but these are some of the theories I have.
They tend to ebb and shift depending on new information provided by Big Finish and the like (I didn't even begin thinking about the Valeyard's future history or strangely enough the Cushing Doctor until
Trial of the Valeyard) which for the most part I rather enjoy and accept, but there are a lot of theories and hypotheses which I just can't get behind, like the whole War Doctor thing. That still strikes me as non-canon personally, it just doesn't sit well with me.
