Galois groups and prime divisors in random quadratic sequences

Given a set S = {x(2)+c(1),....,x(2)+c(s)} defined over a field and an infinite sequence gamma of elements of S, one can associate an arboreal representation to gamma, generalising the case of iterating a single polynomial. We study the probability that a random sequence gamma produces a "large-image" representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over Z[t], and we conjecture a similar positive-probability result for suitable sets over Q. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.