To solve the problem, we need to compute the composition of the function (f(x) = 2x + 3) three times starting with (x=1).
Step 1: Compute (f(1))
(f(1) = 2(1) + 3 = 2 + 3 = 5)
Step 2: Compute (f(f(1)) = f(5))
(f(5) = 2(5) + 3 = 10 + 3 = 13)
Step 3: Compute (f(f(f(1))) = f(13))
(f(13) = 2(13) + 3 = 26 + 3 = 29)
Answer: (\boxed{29})

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