We describe a computation of the first 20,000 partial quotients in the regular continued fractions for Euler's constant у = 0.577... and exp(y) = 1.781... . A preliminary step was the calculation of у and ехр(y) to 20,700D. It follows from the continued fractions that, if у or exp(y) is of the form P/Q for integers P and Q, then |Q| > 10^10000.