D. Ornstein’s fundamental work [Adv. Math. 4, 337–352 (1970; Zbl 0197.33502)] expressing a property of an abstract measure-preserving dynamical system that ensures it is isomorphic to a Bernoulli shift (or IID process) and going on to show that Bernoulli systems of equal entropy are isomorphic was initially firmly located in the category of measure-preserving systems and measurable maps between them. For given finite alphabet IID processes there is a natural topological or coding length structure to hand, and Keane and Smorodinsky went on to show that such processes are finitarily isomorphic (meaning that any coordinate of the target system process can be determined by finitely many coordinates of the source process) if they have the same entropy [M. Keane and M. Smorodinsky, Ann. Math. (2) 109, 397–406 (1979; Zbl 0405.28017)]. Despite considerable efforts and results, including a formulation of necessary and sufficient conditions for a process to be finitarily isomorphic to an IID process by D. J. Rudolph [Prog. Math. 10, 1–64 (1981; Zbl 0483.28015)] a full emulation of the Ornstein theory in the finitary category has not emerged, and the present paper in part explains this. Here a finitary factor of an IID process is constructed that is not finitarily isomorphic to an IID process, it is shown that any ergodic system is isomorphic to a process none of whose finitary factors are IID processes, showing in particular that there cannot be a general finitary version of Sinai’s theorem nor can there be a finitary version of the weak Pinsker property.