Complexity and the Big Bang

After a brief review of current scenarios for the resolution and/or avoidance of the Big Bang, an alternative hypothesis is put forward implying an infinite increase in complexity towards the initial singularity. This may result in an effective non-calculability which would present an obstruction to actually reaching the beginning of time. This proposal is motivated by the appearance of certain infinite-dimensional duality symmetries of indefinite Kac--Moody type in attempts to unify gravity with the fundamental matter interactions, and deeply rooted in properties of Einstein's theory.
Comment: Revised version: minor additions and clarifications; references added