We provide an analytical solution for American perpetual compound options, that do not rely on a bivariate or multivariate distribution function. This model is especially applicable for a real sequential investment opportunity, such as a series of drug development, tests and clinical trials, where the project can be cancelled at any time, and where the probability of failure declines over stages of completion. The effect of changing input parameter values can clearly be seen in terms of resulting overall project process volatility, and the effective mark-up factor which justifies continuing with each investment stage. In the base case, the effective markup factor increases as the stage nears completion if the project failure declines, although the absolute threshold of the project value less the remaining stage investment costs declines. This is consistent with the effect of decreases in project value volatility. Other results are not always intuitive, with different signed vegas and chi's for different investment stages and degrees of moneyness. This study appears to be a unique approach, which yields the threshold project value relative to investment costs that justifies investment at each stage, with no timing restrictions.