When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.

One often hears (or at least I often hear) that HH^* corresponds to A-infinity deformations. I am wondering whether there is any reference which works this out precisely. **(EDIT: This seems to be incorrect. See Damien's answer.)**