Which languages are recursive-only languages?
It depends on what you mean by looping construct - there are several types. Infinite loops, iterators - loops that count each item in an array or hash - and general loops like C style
for ( int i = 0; i < 10; i++ )
Wikipedia has a table of support for such constructs by language: Loop system cross reference table
To answer your question fully, Haskell and Scheme are two examples of languages that do not have standard for
loops built in; they are generally done using recursion.
The obvious answer (if it counts as a language) is various types of Assembly languages.
Prolog and other logic programming languages.
As an aside, doesn't this question more or less boil down to programming paradigm? Imperative languages have looping constructs; other do not have.
Edit: Language designed specifically to make you tear your eyes out, like
Functional programming languages (e.g. Haskell, Erlang) generally don't have loops, nor do function-level languages (e.g. FP, J) or logic languages (e.g. Prolog, Planner). Indeed pretty much the entire group of declarative languages (of which functional, function-level, logic, etc. are a subset) tend not to have looping constructs.
But...
That being said a lot of those have ways of doing much the same as explicit looping. Common Lisp, for example, has macros that give you the ability to do what looks like regular for, while, etc. loops by macro trickery behind the scenes. Dylan (a very un-Lisplike Lisp) goes a step farther and elevates such macros into something that is effectively part of the language (although the semantics can still be defined in terms of recursion and macros). Too, common operations in functional languages like zips, maps, folds, takes, etc. are higher-level functions that mask explicit recursion behind a function call and act in many ways like assorted loop constructs.