Can a button control a 5 speed fan without a microcontroller?
In addition to Chris Stratton's answer, it is (in principle) easy to build a sequencer from a single IC, such as a 74HC163 binary counter. But the heart of the circuit is the least of your issues. First, you would need to debounce a mechanical pushbutton in order to get reliable operation. Second, you would need to decode the outputs of the counter to do the actual control of the fan motor. I've no experience with 5-speed fans, but I suspect this would be the biggest part of the project.