This is unlikely. While the canon status of the Doctor's limited number of regenerations is sufficiently problematic that it is unclear whether time lords do in fact still have a limited number of regenerations in canon (it's never been mentioned in new Who), it was certainly used as a significant plot point in the "Trial of a Time Lord" arc in Season 23 (the Valeyard's reward for successfully prosecuting the Sixth Doctor would have been to receive his seven remaining regenerations) and does form a clear part of the Doctor's complete history.
However, when the War Doctor is included, Matt Smith represents The Doctor's 11th regeneration, so Peter Capaldi is likely to be the 13th incarnation of the Doctor. This would make him the final one according to the classic model, although as we've seen there does appear to be at least one mechanism for adding new regenerations.
However, there is strong evidence in Day of the Doctor that the Doctor will have re-incarnations after the Capaldi Doctor. (Spoilers below.)
In Tom Baker's cameo appearance as the museum curator at the end of the episode, he suggests that he is "trying on an old face". This strongly implies, in concert with the other dialogue in this scene, that the curator is in fact a regeneration of the Doctor from after the Capaldi doctor, and that the Doctor's story does not therefore end with him.
(May I just take a parenthetical moment to curse Stephen Moffat for making the numbering of Doctors more confused, even as I applaud his creativity.)
EDITED TO ADD: Following the events of the 2013 Christmas special, The Time of the Doctor, we now have better information as to whether the 12 regeneration limit will be a problem for Capaldi's doctor. Specifically:
The Matt Smith Doctor is the 13th and, so far as he is aware at the start of this episode, final incarnation of The Doctor. In addition to the War Doctor incarnation mentioned above, David Tennant's use of regeneration energy to heal himself in Journey's End also used up a regeneration. However, during the Christmas special, the Doctor is granted an unspecified amount of extra regeneration energy as a reward for having saved Gallifrey from the Daleks in "Day of the Doctor". Capaldi is therefore the Doctor's 13th regeneration, one beyond the canonical limit, and there are an unspecified number of regenerations left to him.