3
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Mathematica, period 1010108.567841344 9999! ≈ 101010102.192984439
Print[StringJoin[ToString[#0, InputForm], "[", ToString[#1 - 1 /. 0 -> "9^9^99!"], "]"]] & [9^9^99!]
The iterations replace the final 9^9^99! with the literal numbers 9999! - 1, 9999! - 2, 9999! - 3, ..., 3, 2, 1, and back to 9^...