12. If we wish to answer the question as it is asked here, then the correct answer is not 13. We have been told three specific things: First, one of the politicians had correctly stated the total number of lies told before he made his claim. This has to be the first politician, since all others respond with an n + 1 statement (based on the preceding politician's stated number). So, the first lie was already told beforehand, and the first politician's statement was true. Secondly, by the logic I have just described, we have observed the subsequent twelve politicians tell lies. Thirdly, the question asked of us is to determine how many lies were altogether told by these politicians. We cannot infer or assume that the first lie was told by the politicians themselves. They could be discussing a lie that the moderator brought up as an anecdote. So, unless it is explicitly stated they told this lie, it cannot be counted. (The default value is false.)
I may be splitting hairs with this level of precision, but this is the logically correct answer to the question. If this is not the intended answer, then the question should be rephrased.
Edit: To extend the flaw a bit further, it doesn't explicitly state that these were the only possible lies. What about the preceding 30 minutes? They could have told hundreds. The riddle should be reworded.