Probably nobody is surprised that I'd select
Mars Matrix: Hyper Solid Shooting. Specifically the Dreamcast version. I'm probably not super qualified, as I'm sure most of this forum has played more games than me, but at the same time it's hard to imagine something being more appealing to me.
[warning: a huge block of text approaches!!]
I love the depth that the game has in spite of an incredibly simple premise. I love the way that you control 4 weapons with just one button. I like that you score all of your points in stage 6 (pretty much), but all of the chaining happens in earlier stages (because chaining just increases the multiplier).
The pacing (in spite of some weird challenge distribution) is pretty much perfect for me, because the first 3 stages go by in a flash, and increase gradually in difficulty (although stage 3 gets a lot harder when trying to full chain); I ended up being pretty committed by the time I figured out how to get past them, and had to contend with the longest and most difficult stage (4), then after completing that It felt like it would only be a matter of time till I actually had the clear. Plus, 18ish minutes is about right for a game, lengthwise, IMO.
The gold cubes and shield mechanic are really genius (this is really where the coexisting simplicity and depth originates from). It's easy to understand what you have to do, and what you have to do to extend your chain, but there are like infinite different ways you could probably go about it. I like that you can use the shield as many times as you want unlike most bomb mechanics. I like that you have a very flexible shield gauge, so you have control over how long you can use it / how long it will take to recharge, but you can't spam it because you can only use it when fully charged (seems like the perfect way to do a shield, or really any of the manifold "get out of danger" mechanics in shmups, at least for me).
I also love how hardcore it is. Only two extends in the entire length, and most people quit after stage two because they can't handle it. It makes me feel like an internet badass that I ALL'd it

I don't know that I actually agree with it being as difficult as most people think, though. I mean I really am awful at every other game, so yeah... On a related note, it's as fair as any japanese designed arcade shooter with the possible exception of the final phase of the stage 5 boss, which as far as I can tell is RNG and too fast to react to, so you kinda need to get lucky or burn a few of the 5 total lives you're alotted during the course of the game, but I might just suck at dodging. It's all (probably including that boss) pretty well telegraphed AFAIR, but does realistically require memo due to speed.
I even like it Æsthetically, though most people don't. It's not so much that the music or art is good, but the music fits the mood perfectly, the color choice is much less offensive than a lot of shmups, and the explosions and weapons, and masses of gold cubes are extremely visceral, as are the obscenely high scores (though it's not exactly in Giga Wing 2 territory). Also the Forbidden Fruit music is really good, and the ship design and evolutions are cool (as are a few of the bosses, like the crab boss, the rose, etc.)
I like the Dreamcast version in particular because of the score attack mode (which is a lot of fun in its own right, as well as being a decent practice mode), the arrange mode (which I haven't even scratched the surface of, but seems pretty sick), and the fact that they took out the wimpy safe spot on the final boss (h4rdc0re!!!), getting rid of the counterstop (and also increasing the score potential to even sillier levels). I admit to also using the wimpy piercing cannon autofire

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Finally, it's a game that's almost entirely static, but somehow I still really enjoy it every time I play, even now after putting god knows how many hours into it, and having a route that I do the same every time. That's a sign of good design right there!
EDIT: also reading this thread makes me want to try lumines again (have the 360 version)