String field theories, especially open-closed SFT, seems to demand a (super)potential or action involving terms of higher order corresponding to n-string correlators which can not be obtained by iterating 3-string correlators. The corresponding algebra is, at least in part, that known as an $A_\infty$- algebra, which I originally invented for the study of based loop spaces. The talk will present 1) various historical ways of composing strings, each of which has played a role in open SFT 2) $A_\infty$- algebra in its own rite 3) applications to OSFT and O-CSFT Time permitting, relations with the BV formalism will be sketched.