Is it true that The Doctor can only regenerate 12 times?

That rule comes from one story (if I recall correctly "The Deadly Assassin"). But it looks like the writers will change it if needed: http://www.radiotimes.com/blogs/1062-doctor-who-regenerate-immortal-12-times-13-lives/

This is Doctor Who we're talking about. These are the retcon experts. Asking Doctor Who to be internally consistent is equivalent to expecting Douglas Adams to be logical. Literally in some cases, since Douglas Adams wrote for Doctor Who for a while.

Believe me, if they wish to continue past Doctor 13, they will find a way. And it will make some sort of absurd, twisted sense. Or not, depending on the mood of the writer at the time.

There have been three official canon answers:

Doctor Who Season 6 (1969), "The War Games": Indefinitely, "barring accidents".
Doctor Who Season 14 (1976), "The Deadly Assassin": 13.
The Sarah Jane Adventures Series 4 (2010), "Death of the Doctor": 507.

Here's what is supposedly the official word. From Doctor Who is now immortal, reveals the BBC: Edict that Time Lords can regenerate only 12 times before they die has been quietly ditched:

He travels through time and space, saves the Earth, and has millions of fans all over the world. But as every "Whovian" knows, the Doctor cannot last for ever: Time Lords are able to regenerate only 12 times before they die.

Fans have always thought that the 13th doctor would be the last, thanks to a 1976 Doctor Who episode, The Deadly Assassin, featuring Tom Baker as the Doctor in his fourth incarnation, and revealing for the first time the regeneration limit. But a passing comment in a children's television programme later this month is set to rewrite history and cast the Doctor, iconic hero of the world's most successful and longest-running science fiction series, as immortal.

In the Sarah Jane Adventures (yes, that was the old friend), in the episode "The Death of the Doctor". The Doctor is asked how many times he can regenerate, and replies

Five hundred and seven

Recently (always considering that in the Whoniverse the word "recently" is very flexible), Moffat stated he had a plan to skirt the regeneration issue.

The average Time Lord can only regenerate 12 times - this has been pretty clearly stipulated in stories such as The Deadly Assassin and Mawdryn Undead, from the 70s and the 80s... no matter what the Doctor may flippantly say in, e.g., Death of the Doctor.

However, there are certainly precedents for cheating death beyond the 12th regeneration. The Master, in The Keeper of Traken, was in his final regeneration and resembled a decomposing corpse, but managed to steal the body of Nyssa's father Tremas and use a power called "the Source" to enter into a new cycle of regenerations. A few decades on, he's still going strong, in several new bodies.

It seems clearly that if the Doctor Who franchise is still going strong while the 13th actor is in the lead role, they will find some way of giving our hero his own new cycle of regenerations. It's Doctor Who - anything can, and usually does, happen!

It has come up from time to time that the 12 regeneration limit may be some artificial, Time Lord Council imposed thing.

eg. In the Five Doctors, the council convinces the Master to help them out by offering to give him regeneration, as he had no regeneration energy left and was in the Trakenite body of Nyssa's father, Tremas.

From the wikipedia article:

"This is the first time it is suggested that a new cycle of regenerations can be bestowed on a person (in this case the Master), implying that it could be possible to circumvent the twelve-regeneration limit established in The Deadly Assassin."

Based on the recent episode "Let's Kill Hitler", we see River Song giving her remaining regeneration energy to save the Doctor's life, so that may have prolonged the regeneration limit. That's very plausible, considering in "Silence in the Libray"/"Forest of the Dead", River Song physically died. We now know that she can regenerate, but she didn't/wasn't able to. I believe, IMHO, she couldn't have regenerated because she gave it to the Doctor.

In the latest series episode 'Let's kill Hitler', River poisoned him and he went into the Tardis and it said 'regeneration disabled'. I presumed that meant that he ran out of regenerations. And when River was in the hospital, I thought Amy said that she had used up ALL remaining regenerations to save him, not just the last of her energy. So I was under the impression that both River and the Doctor have no more regenerations left. But I'm probably wrong.

No one's mentioned Trial of a Time Lord, where the Master says that somewhere between around the "12th and final regeneration", the Doctor becomes the Valeyard and will seek to undo everything after Colin Baker.

In the audiobook "Ring Of Steel," the Doctor makes a comment about only being young twelve times.

Originally, the doctor has only 12 regenerations, but in the episode let's kill hitler, melody, who has already regenerated twice, gives the the doctor her 10 remaining regenerations, leving him with 22. He then gives her one when she breaks her arm, leaving him with 21 regenerations, 22 lives, and 11 more Doctors. That means we're only half way done with the doctor. At this rate, the show will end in 2063 at least, and by that time we may have ourselves a real doctor!

The 12 regenerations is an artificial limitation probably created by elder Time Lord hierarchies, who knows how long ago. Just watch The Five Doctors, in this story the Time Lords offer The Master a new set of regenerations.

Also, if he could only have 13 incarnations the Doctor would more than likely dire a very young man. The Time Lord lifespan is around 7000 years. The Doctor as far as we know is only around 1,100 years old, because of his lifestyle he's running through his regenerations at an astronomical pace. This always makes me wonder why in the world would NuWho Doctors always claim to be so old. Yes, to humans and most aliens he's old, but to his own people and what he sees as normal for Time Lords, he's a young'un.

I was always under the impression that The Doctor would have 12 regenerations. Like the hours on a clock. Hence *time*lord.

River did give him the rest of her regeneration energy because she's going to the library anyway but not just that he told clyde that he has unlimited the show is so poplar they'll keep him going the doctor is a little energizer bunny he'll keep on going.

According to the official Doctor Who Magazine, which I bought right after they announced Matt Smith, Timelords can only surpass the 12 regenerations if they do the time lord council a favor, and since the tenth doctor killed them all, he's out of luck. They probably will pull the "River gave up her regenerations" thing, but she did so to counter the poison.

It is 13 regenerations. I got a book for Christmas, as I am completely obsessed. It says there can only be 13 but it is possible one can be forged through artificial means. However, these artificial means are most likely only found on guess which destroyed planet. I'm sure Moffat will come up with something, though, he always does. By the way, the fact of regeneration was not even there until Hartnell couldn't do the show anymore, by then the show was way too popular to cancel, they didn't think much of it when it first started, but then regeneration was an option. They made the 13 rule because anyone saying that the show would last 50 years would be put in a madhouse.

First, we should establish the count of used regenerations. In the episode The Time of the Doctor, it's confirmed that the Doctor's Meta-Crisis regeneration did count as a regeneration. Between this regeneration, and the recently-revealed (Night of the Doctor/Day of the Doctor) 'War Doctor' regeneration, Matt Smith, who we previously called the 11th Doctor, was in fact the 13th regeneration of that character. The math works out that the original Doctor was his original face, so the 2nd Doctor is the 1st regeneration and so on. This then means that the 13th Doctor is his 12th and final natural regeneration.

However, just as they did for The Master, the Time Lords granted The Doctor a fresh set of regenerations (The Time of the Doctor). So as the '11th' regenerates, he turns into Peter Capaldi, the first of a new set of 12 regenerations. As of The Time of the Doctor, the Doctor now has 11 more regenerations left.

Largely copied from my answer on Movies & TV.SE

To add to Ross's answer, there's this little exchange from Time of the Doctor:

The Doctor: Everything ends.

Clara: Except you. You pop right back up with a new face.

The Doctor: No, not forever. I can change 12 times, thirteen versions of me. Thirteen silly Doctors.

Clara: But you're the 11th Doctor

The Doctor: Are we forgetting Captain Grumpy? Eh? Now I didn't call myself the Doctor during the Time War, but it was still a regeneration.

Clara: Okay, so you're No. 12

The Doctor: Well No. 10 once regenerated and kept the same face. I had vanity issues at the time. 12 regenerations, Clara... I can't ever do it again. This is where I end up, this face... this version of me.

That was before he got his new regeneration cycle, so we don't really know how many he has this time.

In Series 12, this has been (apparently) retconned.