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][1] by Javier Fresán and Peter Jossen). So the exponential Period Conjecture provides a heuristic coming from algebraic geometry that $e$ is not a period.


  [1]: https://people.math.ethz.ch/~jfresan/expmot.pdf