Summarizing the commentary of Will Brian and myself, there is a notion of IP set (https://en.m.wikipedia.org/wiki/IP_set) (edit: Will says "has your summation property", which I misinterpreted to mean the following ) which has the property that Dominic points out: for every positive integer $n$ there is a member (in fact an infinite subset of them) $z_n$ of the IP set which is the sum in at least $n$ different ways of the members of the base set $A$ which generates the IP set.

Dominic wants a thin or sparse set $A$ to build an IP set. The Fibonacci sequence F provides an example of a sparse $A$ with Dominic's property, as every sufficiently large member of F itself has multiple representations as sums of Fibonacci numbers. However F is not an IP set ( as an infinite IP set has a partial closure property: given a in the IP set there are infinitely many b in the IP set with a+b also in the IP set). Will mentions the example of base 10 numbers having only ones or zeros in their decimal expansion as a sparse IP set.

One can make A, S(A), and further iterations of S on A simultaneously sparse with a modification of the following technique: let A have 1, and for each positive integer n, throw in c=100^{100^n} and c-a for all a put in at an earlier stage. Then A and S(A) have small but widely separated dense clumps of integers, and one can arrange further iterates to be sparse.

Gerhard "And That Ends The Commentary" Paseman, 2017.04.24.