06 Hummer H3 random exhaust smoke

Sounds like either the piston oil rings or the cylinder head valve guides are beginning to fail and allow oil to mix in with the air and gas, hence the smoking exhaust. Try adding Wynn's Engine Restore to your oil plus and Oil Treament, as short-term fixes. At some point you'll need to rebuild or replace that engine.

it isn't burning oil or coolant, that is what I cannot figure out.

If its smoking it's either burning oil or anti-freeze, there is nothing else it could be burning and then releasing from the exhaust pipe, sorry. Now it could be moisture in the exhaust that's evaporating as the exhaust slows heats up and vaporizes any moisture. Driving it would then remove any moisture from the exhaust and then it won't do it again. Common for vehicles that are not driven much. These vehicles tend to have the exhaust system rot form the inside out due to the build-up of moisture and rusting. Burning anti-freeze is a white exhaust smoke with a sweet or sticky odor. Burning oil has a grey/black/dark and heavy exhaust. Leaking brake fluid would have smoke pouring out from the front engine compartment, unless theres a leak in a brake line somewhere. Get it properly checked out.

my coolant level isn't really changing beyond normal. it is white smoke, could it be a small leak in the gasket? it is definitely not oil.

A coolant leak will emit white smoke and smell sickly sweet, an oil leak will emit a grey smoke and smell heavy to the nose. Measure and track your fluid levels carefully each time the same way from the same spot, then monitor them for a few weeks to see if there's any change. Even a small coolant leak will show after a long period of time. But if the exhaust is white, or foggy, and has no foul odor, then it's probably just moisture from inside the exhaust getting heated-up and turning into condensation. This is normal for a vehicle stored after use. Cooling metal will sweat and the surrounding air will release moisture and build on the metal, just the same way a window fogs-up and is sometimes inside a house when the outside is cold or very humid. A leaking head gasket must be caught before it causes permanent damage. Try performing a few cylinder compression leak tests over a long period of time and track the results. Early 90's Chevrolet Cavaliers had defective metal shim head gaskets that leaked coolant into only 1 cylinder head. Sometimes they seal themselves over time and the head and block surfaces warp and mate better, and sometimes they leak more. In cases like this there's no immediate solution but to watch, monitor and track the results over time and then you've got something to help you identify the real problem if there is one. Ever vehicle I've ever owned did this same white exhaust at times. In every case except 1 it's just been moisture in the exhaust system, with 1 case being the head gasket. In 1 different case the valve cover gasket was leaking oil down the back of the engine block and sometimes would end-up on the exhaust, but again, oil produces a thick grey smoke. Good luck.