Is there an explicit example of a complex number which is not a period? Kontsevich and Zagier proposed a definition of a period (see for example http://en.wikipedia.org/wiki/Ring_of_periods ). The set of periods is countable, so not all of $\mathbb{C}$. I heard a rumour today that there is now a known explicit complex number which is not a period; does anyone know if this is true, or have more details?
 A: I recently attended (and blogged about) a colloquium on precisely this topic in which your 'rumour' was mentioned.  Apparently it has been conjectured that neither $e$ nor the Euler constant $\gamma$ are periods.
A: Such a number is constructed in this article of M.Yoshinaga where it is proved that periods can be effectively approximated by elementary rational Cauchy sequences.
A: I haven't heard the latest about periods, but I want to point out a potential fallacy here.  It's said very often that the easy proof of the existence of transcendental numbers (on cardinality grounds) is non-constructive.  But, that's false!  It is constructive.  Given pen, paper and lots of time, I could extract from that argument the decimal expansion $0.a_1 a_2 \ldots$ of the transcendental number that the proof constructs.
See, for example, these comments of Joel David Hamkins.
I suspect that the same is true for periods: that there's an effective enumeration of them, so there's an algorithm for generating the decimal digits of a number that isn't a period. 
