For a static scene, I'd expect the results to have DOF almost as deep as at the tightest aperture (you do record details at that aperture, but they are somewhat blurred by wider apertures - reducing contrast), with some camera shake mixed in. Indeed, this would give an effect very similar to just using an in-between aperture - not very exciting per se.

To have the method resulting in something more unique, something should change in your scene while you're changing the aperture. For example, you could fire key light with one aperture and fill light with other, making areas lit by different lights have different depth of field. In following example, note how the subject is fully sharp (the rear part would be blurred on a wide aperture shot), while the background is blurred (it would be much sharper on a narrow-aperture shot):

Single exposure, 58mm f/2 lens; focus on lettering of the other 58mm lens. Exposure started with f/11, key light LumoPro LP120 at 1/2 power snooted almost top-down; aperture changed to f/2, fill light LumoPro LP120 at 1/16 power diffused at left of scene.

For comparison, here's the same image taken with only wide aperture...

Single exposure at f/2. Key light at 1/32 power; fill light at 1/16 power.

...and only with narrow aperture:

Single exposure at f/11. Key light at 1/2 power; fill light at 1/1 power.

To reduce camera shake and allow for unexposed scene adjustment, you could shoot the different apertures as separate frames and combine them in post. Combined with repositioning an object, this could result in image where the object is apparently blurrier closer to focus plane and sharper further away.