So, as MooseSmuggler mentioned in his spoiler, it would appear confirmed that the Doctor has already run out of regenerations; in addition to the one into the War Doctor that we recently learned of, apparently the aborted regeneration in The Stolen Earth also counted.
Here's an official source, in case anyone wants to be sure that this isn't all just fan speculation (because I wasn't sure for a while depsite having heard rumours).
I don't personally feel like this actually changes that much about the Doctor's imminent visit to Trenzalore compared to if he'd still had one left. It was already clear that he
outright dies on Trenzalore, and so even if he'd been able to regenerate, apparently he was just going to be killed too quickly before he could start to do so. Then again, maybe the only reason he's supposed to outright die on Trenzalore is
because he can't regenerate, and we just didn't know that beforehand. Time travel! It is confusing.
This does at least mean that not only will the Doctor have to find a way to trick out time so that he avoids his death on Trenzalore without causing a paradox with the fact that he supposedly does die there (and leave behind the timestream thingy without which he would not have met Clara) - he'll also have to do so in a way which involves him breaking the rules and procuring himself a new set of regenerations, despite that the Time Lords are currently still lost.
But, Trenzalore aside, knowing that the Eleventh Doctor has actually been unable to regenerate the entire time puts a lot of things from past episodes in a new light, which I find interesting to think about.
There's the time at Lake Silencio where he began to regenerate before being shot one last time - but actually that was the Teselecta putting on a light show for the benefit of the watching companions and Silence who wouldn't have known that he couldn't regenerate and so would have expected to see that. So that remains nicely consistent both with what we thought we knew then and what we actually know now.
Then I can't help but think of the voice interface scene in Let's Kill Hitler, in which the Doctor asks the voice interface "so, basically, better regenerate?", which seems odd if he knew he couldn't - though we can probably put that down to him not thinking straight because he's not really thinking straight for the entirety of that scene. Also interesting is that the voice interface phrases its answer "Regeneration disabled", rather than something like "You have no more regenerations", or maybe "You cannot regenerate", if Steven Moffat wanted to remain more ambiguous. Presumably the poison of the Judas tree actually does disable regeneration, because it'd be silly to try and kill the Doctor with something that doesn't - but "Regeneration disabled" could also perhaps be referring to the external limit put on regenerations by the Time Lords, and it's not so much that they give each Time Lord twelve regenerations to use up, but rather they put some kind of "lock" on every regeneration after the twelfth to disable it from occurring even though it should be able to.
I also wonder about the time on Apalapucia where the Doctor claims that Chen7, the disease that affects two-hearted species, would kill him, "and no regeneration". The implication there is that Chen7 itself disables regeneration, but he words it in a very specific way that doesn't actually link the lack of regenerating to Chen7 and could just as easily mean that he simply couldn't regenerate anyway.
Then his threat to the Cyberplanner of regenerating in order to burn out any Cyber-stuff in his brain could easily have been a massive bluff; he merely "allowed" the Cyberplanner some of his memories on regeneration, so he would have been able to hold back the fact that he'd already used up all twelve.
Regardless of these moments when regeneration itself has been brought up, it's also just quite something to think that the Eleventh Doctor has known he's had no more regenerations left for his entire life. Being him, he'd have tried to ignore it and forget about it (just as he was trying to forget about the War Doctor's very existence), but it'd always have been there in the back of his mind, the knowledge that if he gets himself mortally wounded
there is nothing he can do and he will just plain die. It feels very appropriate now that fear of death - outright death, not regeneration - has been such a recurring theme throughout the Eleventh Doctor's run. It's probably that constant lingering fear of his mortality that's made him so especially good at coming up with last-minute jammy escape plans to avoid his seemingly inevitable death. So I guess it's also fitting that its down to this particular incarnation of the Doctor to come up with some last-minute jammy plan to escape this mortality altogether and get himself more regenerations, even though that's supposed to be impossible. If any Doctor can manage that, it's this Doctor.