Strong exponential integrability of martingales with increments bounded by a sequence of numbers

A strong exponential bound for the probability tail of a martingale whose increments are uniformly bounded by a sequence of numbers is established. The exponent, which grows to infinity faster than a quadratic polynomial, is a Young function determined by that sequence. The related Orlicz norm of a martingale is estimated, some examples are discussed, and the optimality of the exponent is proven.