Let $X$ be a random variable with moment generating function $e^{P(t)}$ for a power series $P(t)$. Assume the moment generating function exists in a neighbourhood of 0. Then the cumulant generating function is $P(t)$ whose coefficients give the cumulants.
My question is, is it true for any distribution apart from the Normal that there are infinitely many non-zero even order cumulants?
A theorem of J. Marcinkiewicz ( "Sur une propriété de la loi de Gauß", Math. Z. 44 (1939), no. 1, 612–618, cited by Linnik, Kagan, and Rao, p. 25, or by Luckacs, p. 213; I have not studied the original paper) says that if a cumulant generating function is a polynomial, it is a polynomial of degree at most $2$. If your $P(t)$ had finitely many even Taylor coefficients, then $P(t)+P(-t)$ would be a polynomial, and hence of degree at most $2$. But $P(t)+P(-t)$ is the cumulant generating function of the difference $X_1-X_2$ of two iid copies of $X$. Hence $X_1-X_2$ is normally distributed, and by a theorem of Cramer, so are the summands $X_i$.