Similar post to Can Arduino control 300+ Relays ? - Frequently-Asked Questions - Arduino Forum. Wondering what kind of system needs a sequence of 300 electromagnets? Sounds like a small EMALS.
With assumptions only one relay need be on at one point in time and a directional diode was used on each. And given the tri-state ability of the output ports. In theory, it seems 18 pins would provide the ability to drive 306 small relays directly in a Charlieplexing like format. Assuming I did the math correctly where the sum of the sequence 2x(n-1) with n limits from 1 to 18 is 306. A simple small example using n=3 pins to drive ((2x0)+(2x1)+(2x2))= 6 relays would be...
This is a lot of relays! And a lot of un-answered questions. This method would probably need a directIO to port library like the one from mmarchetti to get any decent sequence speed. And with one on at a time the sequence speed would be limited by the mechanical close rate of the relays.