Doctor Who: The Atrocity of Mercy
I wouldn't bet any money on it, but I have a feeling that Matt Smith is more than halfway done as the Doctor, and that a new Time Lord will debut either this season or after a series of specials following it like his predecessor. The evidence lies in his actions in this episode.
On a trip to take Amy and Rory to see a Dia de los Muertos celebration in Mexico during the 19th century, the TARDIS comes up 200 miles short of the Mexican border in what we can safely assume is a town in Texas called Mercy. Believe it or not, a TV episode of Doctor Who has not taken place in the Old West since the First Doctor visited the gunfight at the O.K. Corral. The Tenth Doctor did visit the Old West in a novel alongside Martha Jones, and the Sixth Doctor did so as well in an audio story... lending even more weight to my theory that that incarnation met Doc Brown in 1885.
The small town is ringed by a barrier of stones, and the townsfolk are wary of strangers. The Doctor, as always, is simply delighted at the coming adventure, and wastes no time trying to pull off a tough cowboy act lifted from westerns including a really horrible American accent. It made me miss the once-a-series joke that Tennant got where his new companion would try to talk in a time-specific regional dialect to sad effect with him quietly asking them to not do that.
As he introduces himself as the Doctor, Matt Smith is immediately picked up by the people of the town and hurled across the stone barrier as a giant cybernetic gunslinger approaches armed with high tech weaponry. Luckily, the town marshal Isaac relents, and allows the doctor back inside. In the town jail, Isaac explains that the town is called Mercy because of its second-chance philosophy, clearly an enlightened man in an unenlightened time and place.
Turns out the doctor the gunslinger wants is not the Doctor, but a crashed humanoid alien named Kahler Jex. Since his arrival he's served as doctor to the town, saving them from an outbreak of cholera and even hooking up electric heating and street light ten years before their invention.
Brief aside, the year of this episode is debatable. The first electric streetlights were invented in 1875, but Isaac refers to the American Civil War as having ended seven years ago. That would make the year 1872, which likely the answer as we can probably trust a contemporary to remember the Civil War better than the Doctor remembering the exact year electric street lights were invented. End aside.