I don't want anything in my bass case (hard shell) that can move around and damage the finish on the bass. As a result, the following are the only things other than the bass:
1) Strap
2) Clip-on tuner secured in the storage compartment, or clipped onto the headstock if there's no storage compartment.
I seldom use gig bags for any bass other than my Bronco, which is not a good fit for my other hard cases.
I carry the other every-time use stuff - earplugs, small tool kit, circuit checker, guitar cables - in the soft case for my pedal board. The pedal board has three units mounted: a Korg Pitchblack tuner, Bad Monkey OD (modified) and '63 Fender Reverb pedal.
And I have a separate bag for occasional use stuff like gaffer tape, extra speaker cables, mic cables, guitar cables, AC cords, pocket digital recorder, and anything specific to the occasion. This bag often stays in the car unless I need it.