I'm curious. Is it possible to obtain a gold badge without getting any bronze or silver badges? It doesn't seem like it would be.
Yes, mathematically possible but highly unlikely. Almost all of the gold badges are achieved after passing milestones that trigger bronze / silver anyway.
the only possible gold-only badge is the:
not even the Sheriff badges (as the sheriff is only awarded to elected mods, and you need a minimum rep to run in elections, meaning you've probably had a fair number of badges by then anyway).
Unlikely a moderator would not ask, answer, vote or visit for a whole year though. they'd be a pretty crappy moderator if so.
No. Unlike a couple of the bronze badges that are tokens more than rewards, gold badges are, across the board, things you will need to participate and (keyword) contribute towards the community in order to earn.
Gold badges recognize important contributions from members of the community. They are rarely awarded.
On the SE system if you participate in any minimum modicum, you are going to end up with at least one badge to your name.
The only way it could happen is if SE somehow rigs the system: manually awarding a badge by hacking the database or appointing mod status to a dead account.