There is an easy way for them to overcome the 13 regenerations...
For the Season 3 two parter 'Human nature'/'Family of blood', the Doctor puts his consciousness into the fob watch to hide. When he does this, his body becomes human.
In the season 4 finale - 'Journeys end', the meta-crisis produces a second doctor, however his physical form is human and he is left with Rose in the parallel dimension.
So, when the BBC get to the point when the 13th doctor (or 14th, depending on how you look at it) is going to die, they simply have the doctor store his consciousness in the watch again and do the story in a way that the other dimension Doctor is involved. Then he opens the watch and there it is.
Because it's sci-fi, you can say that because it is a new body, it resets the regenerations so he has 13 lives again.
If David Tennant didn't want to come back long(ish) term, you could even say that the strain of taking on the timelord consciousness means he has to regenerate really soon afterwards.
Problems with this is that the other dimension doctor is half human, and the fact that it was specifically stated in journey's end the other universe doctor can not regenerate, and will age as a human. Originally that scene also contained taking a piece of the Tardis and presenting it to him, so he could "grow his own", but it was cut out, Rose and the hybrid Doctor's storyline end there, and interdimensional travel is blocked(again).
There is another doctor clone out there though, Jenny("the doctor's daughter"), who is in her second life. She however does not have a tardis, and the doctor is unaware of her survival, but they might meet again in the future. It's possible that the thirteenth doctor will "retire", and settle down, with a companion of his choice, and present Jenny with the tardis, and graft his knowledge and memories onto her as he did with Donna, but since she is a timelord, she will not have the problems Donna had. No breaking the regenerations rule, a new, female timelord at the Tardis console with a set of regenerations. The actress who played her is the real-life wife of David Tennant, her father Peter Davison(the 5th doctor)
And you are a bit off, you can follow the storyline, but you'll need the "other series", in the "human nature" he opens up his diary, and sketches all his previous incarnations. Again they are visited in the xmas special "The next doctor", and the opening of season 5. And you'll never get the reverse-thingie of River unless you saw silence in the library. And little known plot details from a single episode, like Rory's military training as a roman centurion, the master dropping his ring after his corpse was burned(funny enough in a similar way as Darth Vader was burned in return of the Jedi), and Queen victoria giving chase on him and to create the Torchwood institute, that existed in both and alternate universes, Rose creating the Immortal Captain Harkness, the rise of madam Kovarian, all references to the brigadeer, it is exactly what they do. Not to mention the Dalek emperor, or Davros, or the alt universe current cybermen