I have a ESP8266-01 and a 74HC595 8-bit shift register. I can drive the shift register using the ESP8266, but the ESP8266 doesn't have a SPI interface. I have a Microchip 23LC1024 that I want to drive using the ESP8266. I was wondering if it is possible to drive a SPI slave by bit banging with the shift register?
Yes, you could do it. As long as you are able to read data back from the memory chip somehow.
However, the HC595 is already on SPI-compatible bus. So you would be bit-banging SPI to shift register acting as IO expander, and then use the IO expander to bit-bang SPI to SPI Flash chip, which makes very little sense.
It would make more sense to use the bit-banged SPI to the HC595 also for the SPI flash, you only need separate chip selects (and data back from flash).