Can a language with some set of higher-order functions like map, fold and filter but without recursion or iteration be Turing complete?
A functional language such as (a fragment of) Agda has:


inductive types (lists, trees, numbers, etc.)
higher-order functions
structural recursion (a generalization of fold and map)


However, such a language is not Turing complete. If we add the fix-point operator, which is a higher-order function

fix_t : (t -&gt; t) -&gt; t


then we can use that to implement general recursion and we have a Turing-complete language. Conversely, in Turing-complete functional language it is possible to implement fix_t, at least in the case t = nat.

So the obstruction to Turing completeness of functional languages is whether they can implement the fixed-point operator. (Partial combinatory algebras from another answer to this question also have the fixed-point operator.)