Title says it all, I actually have about four favorite moments in the game.
First Battle with the Four Fiends member Lich - Not only was he both visually and mechanically different from the rest of the games monsters, I really feel the battle with him is the real beginning of the game's narrative. Up until then you've been helping out villages and kingdoms searching for clues about the crystals and finally a tip off leads you to the bowels of a rather nasty dungeon to face this guy, who is much harder than anything else you have faced up until now and signified to the player that saving the world was going to be an uphill battle.
Class Change - While I knew it was going to happen thanks to playing this game later in it's lifetime, but it still didn't quite prepare me for the awesomness that was the class change and how my worthless thief I've been dragging through this game finally payed off for me. It's really the closest thing you get to a moment of shining light of hope in this game as the game finally gets a bit easier thanks to the extra oomph.
Going to the Flying Fortress - I didn't quite expect this, while airships are such a staple of the series that I hardly consider it "high-tech" going to the Sky Fortress was a different story and really changed my view of FFI's world. Up until now I felt like a typical 'swords and sorcery" fantasy with usual fantasy staples, but here you are in a floating city that is flying due to scientific breakthrough. Here you are getting to fight robots after hours of slaying orcs and zombies. It was a real game changer for me that actually really got me thinking about the world of the game outside of the context of the story, which is something that doesn't happen very often.
Time Travel Loop of DOOM!!! - This also caught me by surprise, mainly cause the story has been so straight forward and as an early NES game, you really don't expect much in terms of story telling twists. It's late in the game, and part of it doesn't make much sense, and it certainly makes the ending a bit fuzzy as well, but it was completely unexpected and one of the elements that I really felt differentiated it from other RPG series of the time like Dragon Quest and Phantasy Star. The twist with Garland and the Chaos Temple also created a nice bookend feeling to the games narrative.
So what are your favorite moments?