Your next door neighbor would trip over because it's hard to see in the rain and then sue you for every penny you have. Just kidding.

Off the top my head, probably nothing would happen. Fresh water, particularly rain water, doesn't conduct electricity that well, so unless something physically got stuck into the little slots of your extension cord, I think it's unlikely that the electrical circuit could be closed just by sitting out in the rain. I don't know if it's the building code, but my garage outlets also have GFI protection, like you see in bathroom outlets, so any short circuit would quickly be shut down.

Actually, if it was a hot, sticky night, I'd sneak over to your house and hook up my window AC to it so I could sleep at night, then unplug it again in the morning.