Good pick, part of the reason why it was my favourite game for about a decade. Mario Galaxy is a truly magical experience. I see it, in a way, as the fulfillment of what Nintendo was attempting to achieve with Super Mario 64.

One Game? Too Hard

Wii Sports, Breath of the Wild, Super Mario Bros. 3, Zelda 1, Super Metroid, Metroid Prime, Super Smash Bros. and Mario Kart DS. 

Brilliantly said. This was actually the main reason I think Galaxy 1 is better than 2, it's celestial backdrops and the slightly more poignant vibe of the music gave it more emotive depth than the blue skies and lighter tunes of 2.

That's it! That's exactly it! I was trying to figure out why Galaxy 2 felt unfulfilling and uncharismatic, now I believe you've put the finger on it - it couldn't let you enjoy the wonders of space depths. At times, even the first one took me out of the illusion - especially that dreadful Toy Time Galaxy. And there were times I could feel the emotive depth coming back to me in Galaxy 2, namely in the Cosmic Cove section.

Having actually built upon Rosalina's backstory in the first one helped a great deal. And I believe that's why she's a fan favorite in the Mario pantheon. The second one did push aside anything remnant of a story, it actually shipped you right into the platforming section without making you feel the excitement of running around spherical worlds and see the stars get lost into the Milky Way.

I sort of see Galaxy 2 as an extension of Galaxy 1 in a similar way to the Star Road section of Super Mario World - those levels are my favourites of the game. While they are separate from the rest of the game, the rest of the game still exists — Star Road is not satisfying outside of the existence of Mario World. My favourite 3D platforming is from Galaxy 2, but Galaxy 1 is where the magic of the universe and the unparalleled story/atmospheric experience is. Combined, SMG1+2 is the best of both worlds.

So even though I do like the levels of 2 mostly better than 1, the experience is significantly less without the experience of 1. If you can only play one of the two, make it Galaxy 1.

If Nintendo makes a Galaxy 3 I really hope they capture everything in one game.

There is definitely a degree of emotion, isolation, and then little pockets of tremendous joy throughout the universe of Super Mario Galaxy. Like Breath of the Wild, I think Super Mario Galaxy remains one of Nintendo’s most significant achievements.

Yeah Galaxy 2 definitely has moments of contemplative beauty, but far less so than the first game for me. And yeah, Rosalina's story helped, as did having a full fledged hub instead of the more generic world map of 2.

Galaxy 2 is still an amazing game, don't get me wrong, but the original just has a special magic. It didn't just inspire my brain and test my hands; it touched my heart.

And as a bit of a memory of how awesome Galaxy is:

I love this game, and another thing I should mention was that I love being able to see some fascinating location not yet travelled to ahead of time. The platforming in this game (and Mario Galaxy 2) are both still unrivalled.

The devs' explanation for getting rid of the pirate logs was that it created a plot hole with the Impact Crater (if the Crater was supposed to be sealed, how did Metroid Prime manage to get out, get captured, escape, and then get back). Problem was that explanation seemingly creates another plothole itself, as now there's no good explanation for how Metroid Prime gets its hands on beam weaponry/missile launchers.

If memory serves, the change was in the PAL versions of the GC game as well.

100% this.

But back to the OP's question, I guess if I have to choose, I am going to go with The Legend of Zelda: Majora's Mask.

I was 15 when that game came out.  Ocarina of Time had just come out 2 years before, and was absolutely amazing.  It just blew my mind.  I couldn't wait to see what the follow up to that game would deliver.

Major's Mask just brought so much new that I never expected, and I loved it.  Being able to play as a Deku, Goron, or Zora?  Just amazing.  I loved all the transformations.  It really brought a bunch of different possibilities to the game.  And I had never played in a world that felt so alive.  The 3 day repeating cycle allowed all the NPCs to have these intricate schedules, that really made the world seem alive.  I think that was the part that really stuck with me.  Yeah, a lot of other games had day/night cycles, but this was a whole other level.

The rest of my list looks something like this:

1. The Legend of Zelda: Majora's Mask
2. The Legend of Zelda: Ocarina of Time
3. Super Mario Galaxy
4. Metroid Prime

After that, it gets really tough to distinguish between games.  Hell, even these few are really tough to choose between.