Monotone cumulants of compactly supported measures need not grow exponentially

For a probability measure μ on ℝ with compact support, Hasebe has asked whether its monotone cumulants κₙ(μ) always satisfy an exponential bound |κₙ(μ)| ≤ Cⁿ. We show that the answer is negative, already for μ = ½(δ₋₁ + δ₁), and that exponential growth is in fact exceptional: for every R > 0 the measures on [−R, R] whose monotone cumulants grow at most exponentially form a meagre set. The key observation is that the generating series Σ κₙ(μ) uⁿ⁺¹ is the iterative logarithm, in the sense of Jabotinsky and Écalle, of φ_μ(u) = G_μ(1/u), where G_μ is the Cauchy transform. By a theorem of Erdős and Jabotinsky its radius of convergence is positive exactly when φ_μ admits a complete real iterate, and classical theorems of Szekeres and Baker on fractional iteration then classify the exceptions. If φ_μ or its compositional inverse is rational or entire, or meromorphic with μ non-centred, only point masses have a convergent monotone cumulant series; for symmetric μ such that φ_μ(√v)² or its inverse is meromorphic, the only further exceptions are the arcsine laws, i.e. the monotone Gaussians. This covers every finitely supported measure with at least two atoms, the semicircle and free Poisson laws, and all non-degenerate uniform laws. The record includes the manuscript and a Python/SymPy script that certifies, in exact rational arithmetic, every identity and numerical value in the paper (21 checks), with a README and the expected output.