I don't understand why you would want to do this, but this might do what you're looking for:
(defn cloj-maps [f n x]
(let [f' (nth (iterate #(comp f %) identity) n)]
((fn deep [y]
(if (vector? y)
(mapv deep y)
(f' y)))
x)))
Sample outputs:
user=> (cloj-maps inc 0 [[1 2] [3 4]])
[[1 2] [3 4]]
user=> (cloj-maps inc 1 [[1 2] [3 4]])
[[2 3] [4 5]]
user=> (cloj-maps inc 2 [[1 2] [3 4]])
[[3 4] [5 6]]
user=> (cloj-maps inc 3 [[1 2] [3 4]])
[[4 5] [6 7]]
user=> (cloj-maps inc 4 [[1 2] [3 4]])
[[5 6] [7 8]]