Define your rule as a function:

 ruleDerF = (
 # /. Derivative[n_][f][var_] :> (D[x f[x], {x, n - 1}] /. x -> var)
 ) &;

 f''[x] // ruleDerF

> f[x] + x Derivative[1][f][x]

 FixedPoint[ruleDerF, f''[x]]
> f[x] + x^2 f[x]

 FixedPoint[ruleDerF, f''[1]]
> 2 f[1]

(*havent tested heavily*)