<p>We compute explicit solutions to some double (two-stage) optimal stopping problems with payoffs associated with the perpetual American compound lookback and integral options with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of either the running maximum/minimum process or the integral process. The original double optimal stopping problems are embedded into the sequences of two single optimal stopping problems for the corresponding two-dimensional continuous Markov processes. We develop the change-of-measure methodology to reduce by one the dimensions of the resulting optimal stopping problems. The latter problems are solved as the equivalent free-boundary problems by using the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the two-dimensional state spaces. We show that the optimal exercise boundaries are determined by means of the explicit solutions to the associated arithmetic equations.</p>