Keeping within the official Doctor Who canon, is it possible that the Doctor could transfer regeneration energy to his past self an unlimited number of times? Is there any aspect of the Who universe that would indicate that this is not possible?
I'm not certain that transferring regenerations between himself at different points in his timeline would work as supported by Who canon. Of course, if a writer felt such a solution was the answer to a problem, everything would be Timey-Wimey and such a feat would be accomplished regardless, but I can't point to any extant material that supports such a scheme.
That said, it is possible to gain more regenerations. The Doctor received a new cycle of regenerations from the Time Lords in The Time of the Doctor after all! Similarly, the High Council of Gallifrey offered a new cycle of regenerations to the Master in The Five Doctors, and the Master claimed in The Sound of the Drums that the Time Lords resurrected him to fight in the Time War, granting him a new cycle of regenerations in the process.
The exact system for granting a new cycle remains unexplained.
You have two glasses. A[ ] B[ ]
You fill one up from the tap. A[=] B[ ]
You wait 5 minutes then travel back in time. pA[=] pB[ ] fA[=] fB[ ]
You pour your full cup into your past self's empty cup. pA[=] pB[=] fA[ ] fB[=]
Where did the extra water come from? Answer: there is no new water, it's the same old water. The water started off in cup A in the past; it then got carried over to cup A in the future. That water was then poured from cup A in the future to cup B in the past. The water is then in cup B where it remains in the future.
At one point during all this, both cups are full; but for both cups to be full a cup in the future must be poured into a cup in the past.
Change the cups for timelords and the water for regeneration energy and you see that it's impossible to create infinite regeneration energy, the energy always has to go somewhere. The transfer is set in stone, any deviation would cause a paradox or some similar time problem.
If you prefer to think in terms of timelines, the energy/water's timeline is a linear piece of thread. All time travel does is fold the thread in on itself; the thread is still the same thread.
Here's a diagram, with time passing from left to right:
Yes If The Master is to be believed. In The Season 12 finale of Doctor Who, “The Timeless Children“, The Master persuades the Doctor to join him on Gallifrey, where he forces her to enter the Matrix. He shows her the secret history of Gallifrey and its native Shobogans.
Tecteun, a space explorer, found a "timeless child" with infinite capacity to regenerate.
She adopted the child and studied her, successfully grafting her regeneration capacity into the Shobogans, transforming them into Time Lords; they chose to limit a Time Lord's regenerations to twelve.
The Master reveals that the Doctor is the "timeless child".
Tecteun and the child were inducted into a clandestine organisation called the Division, the details of which were redacted from the Matrix.
Unless of Course the Master is lying
No, he has a limit. This was addressed in The Deadly Assassin, The Doctor Who movie, and confirmed in The Time of the Doctor. If you rewatch some of the episodes, you'll see that the War Doctor does count as a regeneration (and is actually the true 9th incarnation), then he regenerated into the 9th Doctor (10th incarnation), then he changed into the 10th Doctor (11th incarnation), then he regenerated and almost changed in The Stolen Earth/Journey's End (making him the 12th incarnation, possibly, but with the same face), and then he changed into the 11th Doctor (13th incarnation). In the special, the Doctor mentions that every single regeneration counts, including his Warrior one, and the fact that his 10th/11th incarnation regenerated twice, makes it so that the 11th Doctor was always going to be the face he dies as (hence The Fall of The Eleventh). The Time Lords ended up giving him a new regeneration cycle, proving that he couldn't just give himself regenerations, because if he could, the 10th Doctor might have avoided changing for god knows how long. Even with the new cycle, we're not sure how many regenerations he has now, but he clearly has a limit.
Possibly, spoilers for Doctor Who season 12 below:
If Doctor Who episode "The Timeless Children" is to be believed, according to "The Master" The Doctor was not born on Gallifrey and was instead sent from a "Parallel Dimension" from Somewhere or Somewhen and was discovered by a native inhabitant of Gallifrey in "The Early Days Of Gallifrey" and experimented on to create the ability to Regenerate. But this could all be lies because it was told by The Master and someone who just happened to call themselves The Doctor rather conveniently whilst The Master controls aspects of the Matrix.
So yes if The Master is correct then He/She could infinitely give her/his past incarnations extra regeneration energy but it would be useless because if The Master is telling the truth then The Doctor is immortal.
I feel certain the answer is no - The Doctor could not use time travel to keep giving himself his regeneration energy back - for TWO reasons - one canon, the other is simple mathematics.
In-universe, The Doctor meeting him/herself is in itself an anomaly. When Doctors from other timelines leave, they do not return to their timeline - they do not "retain" the experience. Whatever happens during this anomaly does not return with them to their timeline, and only the remaining Doctor whose timeline it was retains the experience. Note this conversation from The Day of The Doctor:
WAR DOCTOR: I won't remember this, will I?
ELEVEN: The time streams are out of sync. You can't retain it, no.
But even if all that could be ignored, the mathematics of it means it isn't possible.
Ignoring this new part of The Doctor's previous existence as 'The Timeless Child' that we currently know nothing about (or even if it is all true), we must assume that, even if The Doctor did previously have unlimited regeneration power, The Time Lords found a way to limit The Doctor to 12 regenerations (making for 13 lives), perhaps using the same genetic engineering they used on everybody else. If this were not the case then Eleven would not have sat dying of old age, unable to regenerate until a new cycle was bestowed upon him. If The Doctor holds enough energy for 12 regenerations, he could only give his past self whatever was left, so he could still only have 12 regenerations. Also, if a Time Lord can only hold 12 regenerations, it isn't possible to "stock up" on regenerations.
Sixth Doctor has 7 regenerations left. He travels back in time to meet First Doctor. First Doctor is already holding 12, so he can't hold any more.
Sixth Doctor travels forward in time to give Eleven (who is actually Thirteen because of Ten's wasted regeneration and The War Doctor) his remaining 7 regenerations. This causes an anomaly, because no matter how many regenerations he gives Eleven, Six now does not have enough regenerations to ever become Eleven.
Ten travels back to give Nine his remaining regeneration. Nine now has one extra regeneration, but for this to be sustainable, he must do the same when he becomes Ten, so he won't actually get an extra regeneration at all. After he gives the extra back to himself as Ten, he will just have his original 1 (or 2, depending on whether Ten has wasted his regeneration on his hand yet) left.
My short answer is "Yes! Doctor Who is potentially immortal."
Feel free to prove this theory wrong after considering the following:
The Doctor has met and even touched himself without any life threatening problems.
See Has The Doctor ever met himself?
What is true in the Doctor's past is true in his future. In The Day of the Doctor, the War Doctor begins running some calculations on his sonic screwdriver which are completed on the eleventh Doctor's sonic screwdriver. This was apparently a rewrite of history in which the eleventh Doctor's screwdriver had not been running these calculations until that event.
Time Lords can transfer regeneration energy to one another. Melody Pond uses an entire regeneration worth of energy, sacrificing her final regeneration, to save the Doctor's life in Let's Kill Hitler. In The Angels Take Manhattan, the Doctor heals River Song's broken wrist with some portion of regeneration energy. In The Time of the Doctor, Time Lords transfer regeneration energy to the Doctor, allowing him to regenerate into his twelfth incarnation.
The Doctor's Dupe Regeneration Hack
To avoid the paradox of him committing suicide in this process, let's say the Doctor has at least one regeneration left. He goes back in time just ten minutes in his own time line and transfers that regeneration energy to his past self. His past self now has two regenerations available. In theory, the future Doctor will now have two regenerations available as well.
If the future Doctor instantly receives a new regeneration upon transferring it to his past self in the same way that the future sonic screwdriver had been running calculations from the past, he could repeat this process until reaching his full capacity for regenerating energy. Then he could return to his time line, ten minutes in the future, and go about his business.