Defining recursive function as a monad (or other solutions)

Is it possible to create recursive function using monads in a functional language?

definition of monad in a functional language should look like: "a->(a->b)->b". Is it possible to have "a" as a parameter, get "b" as a result and recursively repeat it until some condition is satisfied? I've been strugling over factorial example, but I don't see a clear solution.

The reason I'm asking is that I'm defining a language where functions are passed around together with applied parameters. If I repeatedly call a function by a recursion, new data sums with previous data that was there before the recusion was applied. So finally I'd get a set of all intermediate results, while I'm interested only in the final result. I'm not even sure if monads would work in this environment with data stitched to function calls.

Actually, a solution I can use is first to create a set of parameters of each recursion step, and then to call some kind of fold function over the set that produces a single result. But I want to explore other possibilities, so I'm interested in monadic or other solutions.

Thank you for your time.