What are reasons to believe that e is not a period? In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are. The former claim is repeated at many other internet sources including Wikipedia but nowhere could I find a heuristic making the conjecture that $e$ is not a period more plausible than its negation. Does anyone here know of such an argument?

EDIT: I figured it would be good measure (i.e. 'shows research effort') to write what was the best I could come up with myself. I don't find it very convincing however so feel free to ignore. The number $e$ is more or less defined as the value at a rational number (1) of a function that is a solution to a ordinary differential equation ($y' = y$) with rational boundary condition ($y(0) = 1$). Now K & Z point out that all periods arise in this way (replace a rational number in the defining integral with a parameter and it will satisfy an ODE). However they also warn us that the differential equations are really special and (conjecturally) satisfy a lot of criteria among which having at most regular singularities.
Now the singularity at infinity of $y'= y$ is not regular as it has order 2 (while the equation is of order 1) but of course this proves nothing since nothing is stopping $e$ from being the value at some rational number of a solution to a much more complicated differential equation which might be of the right class. So what is missing from an argument along these lines is some way of making precise that $y'= y$ really is the simplest equation which produces $e$ and that 'therefore' more complicated equations can be 'reduced' to it by a series of simplifications innocent enough to preserve the regularity of the singularities if it exists (quod non). Now personally I would not buy such a claim if it wasn't for the fact that it is a bit akin to conjecture 1 from K & Z. However this line of reasoning requires a lot of 'making precise' and perhaps is an entirely wrong way of looking at it, so better ideas are welcome!
 A: $e$ is the value at a rational number of a single-valued meaningful holomorphic function whereas periods are values of multivalued functions (they have "conjugates").  It would take some work to make this statement really meaningful, though... 
Some further intuition may be found in Yves André's paper "Galois theory, motives and transcendental numbers".
A: Periods arise from the comparison between Betti and de Rham cohomology for an algebraic variety. The Period Conjecture, due to Grothendieck, is a transcendence conjecture for periods which says that every algebraic relation between periods arises from geometry (in a certain precise sense).
More generally, there is a wider class of complex numbers called exponential periods arising from the comparison of rapid decay cohomology and de Rham cohomology. The number $e$ is an example of an exponential period. There is an analogue of the Period Conjecture in the setting of exponential periods, and the truth of this conjecture would imply that $e$ is transcendental over the ring of ordinary periods (see Proposition 10.1.5 of the paper Exponential Motives by Javier Fresán and Peter Jossen). So the exponential Period Conjecture provides a heuristic coming from algebraic geometry that $e$ is not a period.
A: To my understanding, the reason is simple: in the almost 300 years since $e$ was discovered, no representation of it as a period has been found. I think this is quite a strong evidence.
Remark. To those who think that periods were introduced by Kontsevich and Zagier, I recommend the paper of Euler, On highly transcendental quantities which cannot be expressed by integral formulas
English translation.
(Strange that he does not mention his own $e$ as a candidate. Perhaps he was still looking for an integral that equals $e$ when he wrote this paper.) 
A: One very weak heuristic comes from the continued fraction expansion
 $$e=[2;1,2,1,1,4,1,1,6,1,1,\dots],$$
which exhibits a very simple pattern. If you exclude rational numbers and quadratic irrationals, which have finite and periodic continued fractions respectively, the rest are expected to have "generic" continued fractions. In other words you expect their continued fractions to exhibit properties of "almost all" real numbers. One such property is the convergence of the partial geometric means to 
Khinchin's constant.
For example this property seems to be satisfied numerically by $\pi$.
I guess this is in the same vein as the heuristics that non-rational algebraic numbers (or even periods) are normal in every base. 
