Push button for different actions in every press?
It's quite easy to do. You need to debounce the pushbutton and use it to clock a counter/decoder that drives the LEDs, eg. HC4017. Wire it to reset after the last LED. 

Two chips (eg.+ 74HC7002), maybe three if you want a bulletproof reset to a known state at power up. Use one ST Nor to debounce the pushbutton (with an RC and pullup and use the remaining 3 NORs to make an OR gate to allow MR to be actuated with an RC or with a proper reset chip or the 5th output (#4). Drive the LEDs with outputs 1,2,3. Output 0 drives nothing (it represents the 'off' state). 

Or use an 8-pin microcontroller and virtually nothing else.