The iterative conception of set has been defended as a natural and non-arbitrary successor to the inconsistent naive conception, but in ‘The Iterative Conception of Set’ George Boolos showed that the hierarchical picture of the set-theoretic universe given to us by this conception fails to lend support to some of the axioms of ZFC, most notably choice and replacement. Both these axioms are delivered by a rival conception of set—the limitation of size conception—but unhappily this puts the axioms of power set and infinity beyond our reach, and has struck many as merely a technical device designed to avoid the paradoxes, rather than a genuine elucidation of our conception of set. Boolos has suggested that perhaps our conception of set is a hybrid of the leading thoughts behind the iterative conception and limitation of size, and in this paper I begin an assessment of the prospects of such a conception. I argue that even if this hybrid conception—the limitation of iteration conception, as I call it—can deliver all of the axioms of ZFC, it does so only if we are willing to make assumptions justified (if at all) only on pragmatic grounds. Insofar as our project is that of providing conceptual grounds on which to believe the axioms of ZFC, I conclude that we have reason to reject the limitation of iteration conception.