Most memorable first boss?

My most memorable would have to be in Tales of Symphonia's Kratos.
That fight was so feelsy, it burnt my feelsy heart.

 

Queen Gohma from OoT! As a first-time player, she's not easy, nor hard. The perfect difficulty for a first boss, and a good design and environment for a first level/dungeon. This is what (I think) makes her the most memorable first boss in gaming.

 

Ghetsis from Pokémon B/W. Man, that was so good!

 

Fighting Vile at the end of Megaman X's intro stage was powerful, and Egoraptor's Megaman Sequelitis episode captured it perfectly.

Go play Megaman X right now if you haven't, because I'm about to spoil the first 5 minutes, and the game is awesome. Like, really awesome. So, you do the first level, learning to blow up robots by shooting them with your buster cannon, then woop, giant robot in an mech suit comes down and attacks you for no reason. He has no life bar and shows no sign of taking damage (why put a robot in a mech suit, I don't know). He's too tall to jump over and dodge his physical attacks, so you WILL lose in a matter of seconds. He zaps you into an electrocage and grabs you, dialogue box pops up and he calls you a worthless piece of scrap metal (which is exactly how you feel, because you just lost to the first boss, probably not knowing your defeat was unavoidable).

Then the important part comes: your best pal Zero shows up, blasts Vile's mech suit arm off with a single buster shot and sends him running. In dialogue, Zero tells you that as you are, you are weak, but one day, you WILL be as powerful as he is, or even more powerful. He just OHKO'd this guy you couldn't even damage, and then he tells you that you can do that and more before the end of the game! That is a good, motivating first boss fight.

 

I don't get what you mean first. First in the game or first experience with said boss?

Assuming the latter, I think the Henry fight at the end of No More Heroes was memorable on my first run, but even more so was the battle with Giygas from Earthbound.

Earthbound tackles a lot of adult themes such as police brutality, cults and trauma, but in very subtle ways (even Magicant was slightly subtle). As a result, Giygas is a complete contrast to everything else in the game. Sure, it's tackling an adult theme (more trauma, the "big bad" of growing up and rape), but it's not subtle about it at all. Giygas is clearly based off of such themes (just look at his speech pattern "It hurs" "I feel good") and also covers multiple at the same time, yet is very obvious about it unlike anything else in the game, so as a result it sticks out like a sore thumb to a new player. It's a masterpiece that I'm not going to forget for a long time.