### Answer: not known
Many thanks to all who helped refine this question and the definitions associated to it. I have rated [Sasho Nikolov's answer "accepted,"][Sasho's answer] because it provided a formulation that captured the intent of the question: the answer to the question is (apparently) not known.
[Philip White's valuable answer][Philip White] has motivated a careful distinction between incomprehensible TMs and strongly incomprehensible TMs (see below).
Further answers are welcome (needless to say), and quite possibly further definitional tuning is needed. For me one main lesson is that this class of questions is challenging to formulate and even more challenging to answer rigorously.
To the best of my ability, the following (finalized) version of the question incorporates all of the valuable insights and suggestions provided by Tsuyoshi Ito, Marzio De Biasi, Huck Bennett, Ricky Demer, Peter Shor, and also a valuable weblog post by [Luca Trevisan][Luca]. All remaining errors or infelicities are mine alone.
### Formal definition
<i>Incomprehensible Turing machines</i> are defined (within ZFC) as follows:
><b>Definition</b> Given a Turing machine M that provably halts for all input strings, M is called <i>incomprehensible</i> iff the following statement is neither provable nor refutable for at least one positive semidefinite real number $r$:
>> <b>Statement:</b> M's runtime is ${O}(n^r)$ with respect to input length $n$
> Conversely, M is called <i>comprehensible</i> iff it is not incomprehensible.
### Disambiguating *decidable* versus *neither provable nor refutable*
The Wikipedia entry "[Undecidable problem: Examples of undecidable statements][wiki undecidability]" concisely reviews the differing senses of the term "undecidable" that are customary in the roof-theoretic versus computability-theoretic literature. With a view to avoiding ambiguity, this question employs exclusively the terminology "neither provable nor refutable."
Further references in this regard are Jeremy Avigad's course notes "[Incompleteness via the Halting Problem][Avigad]", Scott Aaronson's weblog essay "[Rosser’s Theorem via Turing machines][Aaronson]" and Luca Trevisan's weblog post [Two interesting questions][Luca].
### On the existence of incomprehensible Turing machines
That incomprehensible Turing machines exist follows concretely from [a construction by Emmanuele Viola][Viola] and broadly from the complexity-theoretic framework of Juris Hartmanis. In particular, Viola's construction provides, via the methods of [Jeremy Avigad's course notes][Avigad] (as I understand them), the following lemma:
><b>Lemma [Viola's Implication]</b><br> (if a language L is decided by a comprehensible TM) $\to$<br> (L is decided by an incomprehensible TM).
It is natural to wonder whether the converse implication is true. [Philip White's comment below][Philip White] shows how to trivially reduce incomprehensible TMs to comprehensible TMs via *polylimiters*, which are computational modules that (in effect) "pad" the runtime of an incomprehensible machine so as to reduce it to a comprehensible machine.
We therefore distinguish a class of *strongly incomprehensible TMs* as follows.
### Strongly incomprehensible TMs
We say that a Turing machine M having a lower-bound runtime exponent $r$ is *efficient* if it decides a language L that is decided by no other TM having a runtime exponent smaller than $r$. We say that a language L is *incomprehensible* if it is decided by at least one Turing machine M is that is both efficient and incomprehensible, and moreover there is no efficient and comprehensible TM that provably (in ZFC) decides L. And finally, an incomprehensible TM that decides an incomprehensible language is said to be a *strongly incomprehensible TM*.
Thus every incomprehensible language is decided by at least one TM that is both efficient and strongly incomprehensible.
### The three questions asked
> <b>Q1</b> Does the <a href="http://en.wikipedia.org/wiki/P_%28complexity%29#Definition">complexity class P</a> contain incomprehensible languages?
> <b>Q2</b> Can at least one incomprehensible language be represented concretely? (if so, provide a constructive example).
> <b>Q3</b> Can at least one strongly incomprehensible TM be represented concretely? (if so, provide a constructive example).
To the best of my (decidedly non-expert) knowledge, these are an open questions in complexity theory … definitive references to the literature especially are desired.
----
### Motivation
The strongly incomprehensible properties of the complexity class P, and of the Turing machines that decide the languages of this class, presently obstructs my own understanding of a broad class of problems that includes Terry Tao's [Blue-Eyed Islanders Puzzle][islanders], Dick Lipton and Ken Regan's [Urn-Choice Game][urn], and their hybridization in the context of Newcomb's Paradox via the [Balanced Advantage Newcomb Game][Newcomb].
As Juris Hartmanis' monograph *[Feasible computations and provable complexity properties][Hartmanis]* (1978) puts it:
> Results about the complexity of algorithms change quite radically if we consider only properties of computations which can be proven formally.
The struggle to construct well-posed definitions and postulates that capture Hartmanis' insight has helped me to a better appreciation that the complexity class P has some exceedingly peculiar languages in it, that are recognized by exceedingly peculiar Turing machines, whose properties we are (at present) very far from grasping. It is striking that in a completely rigorous sense, it is not presently known whether the complexity class P is comprehensible.
Many thanks are extended to all who contributed comments and answers.
[Ito]: http://cstheory.stackexchange.com/questions/11570/does-p-contain-languages-recognized-solely-by-incomprehensible-tms#comment31612_11570
[urn]: http://rjlipton.wordpress.com/2012/05/23/beyond-las-vegas-and-monte-carlo-algorithms/#comment-20565
[islanders]: http://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/
[Newcomb]: http://philtcs.wordpress.com/2011/12/04/class-12b-newcombs-paradox-and-free-will/#comment-539
[Viola]: http://cstheory.stackexchange.com/a/5006/1519
[Sasho]: http://cstheory.stackexchange.com/questions/11570/does-p-contain-languages-recognized-solely-by-incomprehensible-tms#comment31620_11570
[Avigad]: http://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf
[Aaronson]: http://www.scottaaronson.com/blog/?p=710
[wiki undecidability]: http://en.wikipedia.org/wiki/Undecidable_problem#Examples_of_undecidable_statements
[Luca]: http://lucatrevisan.wordpress.com/2011/02/04/two-interesting-questions/#comment-2962
[Sasho's answer]: http://cstheory.stackexchange.com/a/11575/1519
[Hartmanis]: http://books.google.com/books?id=DTMrTZJH1NYC&pg=PA41
[Philip White]: http://cstheory.stackexchange.com/a/11592/1519