I use a "Maintenance Charger" by ACI, model 1206CC.
One of my vehicles stays plugged into the charger for more than two months

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Yep, we just had to replace a 5 month old battery today. It just started to poop out. Click, click!
It is a platinum 7 year battery, so not a cheap one. These h3's just seem to abuse and use em up.

 

You need to buy a DVOM and hook it up between the negative terminals. Check the amp draw using the mili amp scale. Pull each fuse to see if there are any changes. Note what value they are. Worse comes to worse PULL them ALL...your source for a voltage draw could be anything from the memory of your seats or even a glove box switch that doesn't turn off. Its a hunting game. Remember that when you do this and the door is open it becomes a draw so its best to draw its fuse so that you can check on the others.
You will find it - just takes a methodical routine. When you do find it then look at the wiring schematic to see which circuit it controls. Just because its one fuse doesn't mean it controls only one thing. Good luck.

 

Use caution if you are going to purchase a "trickle charger". Most trickle chargers will continue to charge your battery and if left on long enough will over-charge and damage your battery. As wfturner mentioned, use a maintenance charger.

I use the Battery Tender brand. I have their Plus High Efficiency 12V 1.25 Amp Battery Charger (PN: 022-0185G-DL-WH). The big benefit of a maintainer is the ability of the charger to go into "float mode" once the battery is fully charged. This prevents the battery from being over-charged.

Although you indicated you've replaced the battery twice, you didn't mention how old your current battery is. If it is dying after a short period of time you may have one or more dead cells.

Any aftermarket products installed (stereo, nav, alarm system, phone chargers, winch, etc)? These could also be causing a drain.

I assume your dome light goes out once you close the door (there is a slight delay). You may have a potential door switch issue either with the doors or the tailgate door.

 

My 07 H3 recently started drawing 450mA and that was after disconnecting the onstar fuse. Turned out that my H3 was a lease vehicle when new and the company installed a GPS unit under the passenger seat, under the carpet. Took me hours to find it, just a little black box. It had batteries in it and when they died it was drawing from the battery. Got rid of that GPS unit and all good. Draws about 100mA now. No problems even if it sits for week.

 

mine is 114 ma for years