Shift registers need 3 digital pins in order to work (data, clock and latch), but Attiny45/85 chips provides 2 digital and 2 analog pins.
Would I use 1 analog pin as digital pin? How?
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As Ingnacio has already pointed out, PB0 to PB4 can be used as digital IO ports without problems. It night be a good idea though not to touch PB6, the
Btw, this isn't different from an Arduino, where all so-called analog pins can be used for digital output as well.
If you want to drive shift registers from an ATtiny85 using SPI, you might want to have a look at the tinySPI library written by Jack Christensen. Please note that I haven't used this library personally!