If I had a circuit that required reading the values of four variable resistors (trim pots for example), is there a pin saving way to do this? I know I can do multiple buttons with a shift register but I'm unsure about the trim pots...
Yes, you can use an analog multiplexer.
There is a playground article here: Arduino Playground - 4051
Well talk about fate... I stripped down some old electronics tonight and grabbed a bunch of ICs off a board... one of them is an analog multiplexor 
Well talk about fate...
have fun!