I was reading this article and was wondering if there is any finite set of higher-order functions like map, fold or filter such that they could empower a language to be Turing complete even without having explicit constructs for recursion or iteration. Obviously the higher-order functions would be implemented with recursion or iteration, but the requirement is that recursion or iteration could not be directly exposed to the programmer. That is to say, recursion or iteration can be baked in a finite set of pre-defined, built-in higher-order functions, but the language must not have an explicit construct for iteration or recursion. The set of higher-order functions can be arbitrarily large, but must be finite.
Alternatively, if the answer is no, which is the biggest set of problems that could be computed by such a language? Is that set a known set (eg. equivalent to pushdown automaton)?