We will see there is a well defined limit, in the Grothendieck ring of varieties, of several different sequences of moduli spaces. In flavor, these are algebro-geometric analogs of topological results about homological stability, or of arithmetic results about asymptotics of points counts on moduli spaces. For example, we determine the limit of the motive of smooth divisors (or with s singularities) in increasing multiples of a linear system. We also determine the limit of the motive of configuration spaces of distinct points (or points that are allowed to come together to a limited extent) as the number of points increases. All of these limits in the Grothendieck ring are given by explicit formulas in terms of motivic zeta values. Our results motivate a large number of conjectures in topology and arithmetic. This is joint work with Ravi Vakil.