Throughout mathematics and its applications, one encounters sequences of algebraic varieties-geometric structures defined by polynomial equations. As the dimension of the variety grows, typically so does its complexity, measured, for instance, by the degrees of its defining equations.

And yet, many sequences stabilise in the sense that from some member of the sequence on, all complexity is inherited from the smaller members by applying symmetries. I will present several examples of this, as yet, only partially understood phenomenon. Beautiful combinatorics of well-quasi-ordered sets plays a key role in the proofs.