The elementary real-valued functions are not closed under integration. (Elementary function has a precise definition -- see Risch algorithm in Wikipedia). This means there are elementary functions whose integrals are not elementary. So we can construct a larger class of functions by adjoining all the integrals of elementary functions. You can repeat this process indefinitely. If I understand things correctly, the set of functions that is the countable closure of this process is closed under integration. Does any finite iteration of the process achieve closure under integration?
My guess is no. Has anyone thought about this?