Sine Is Bounded on a Line, and a Line Is Almost None of the Plane
Sine is differentiable everywhere and never leaves the band from minus one to one on the real axis, which makes it the counterexample everyone reaches for, and the modulus of sine at 10i is already 11013.23. The Cauchy estimate caps every Taylor coefficient by M over r to the n on a circle of radius r, so growing r kills every coefficient above the constant one and nothing but a constant survives. The same estimate with a polynomial growth bound gives more: an entire function bounded by C times one plus the modulus of z, all to the k, is a polynomial of degree at most k, and one corollary of that is the fundamental theorem of algebra.