how to calculate the maximum obtainable directivity for a microphone array?
When using a microphone array for baseband acoustic signals, Array Gain is a preferred performance metric over Directivity Index. 

In geophysics, the term “stacking” is often preferred over the equivalent “delay-and-sum beam former. It is certainly more descriptive because you essentially time align signals like a stack and add them coherently.  

If the noise is independent at each microphone, for the aligned stack, the upper bound on coherent gain is $10 Log N$. 

Directivity is a function of the beam pattern which is a function of a single frequency.  For an RF system with an obvious carrier, the pattern at that nominal frequency is an often used way to characterize gain. 

The stacked array gain is baseband. It is the upper bound on gain.