Always do some form of research, even if informal. Five-minute hallway tests are better than zero user input.
Best answers match methods to questions: interviews for "why", surveys for "how many", usability tests for "can they", analytics for "do they". When budget is zero, strong candidates mention guerrilla testing, internal dogfooding, support ticket analysis, and existing analytics. They do not skip research entirely.
Tests resourcefulness and research rigour. Red flag: designers who only do research when given budget and time. Good sign: creative approaches to getting user input with constraints.