For the discrete-time additive white generalized Gaussian noise channel with a generalized input power constraint, with the respective shape and power parameters >= 1, we derive an upper bound on the optimal block error exponent. Explicit asymptotic upper bounds in the limit of a large block length n are given for three special cases: the Laplace noise channel and the Gaussian noise channel with the average absolute value constraint, and for the Laplace noise channel with the second power constraint. The derivation uses the method of types with finite alphabets of sizes depending on the block length n and with the number of types sub-exponential in n.