Man, I really dislike threads where people just list games. WHY do you like this game. Encourage others to try it out or defend it from it's otherwise poor or unknown reputation.
I personally gravitate toward Psikyo and CAVE stuff - not unpopular, but they simply resonate with me.
Psikyo for the randomn start of the first 4 stages - it helps to make each run feel unique. I also really appreciate the steady and noticeable increase in difficulty. Lastly, the stage length feels appropriate. Some shmups have painfully long stages (*cough* Bullet Soul, or occasionally oddly short. Psikyo has the timing down really well.
Gunbird 2 is a favorite. I like the chaining system with medals. Super easy, yet leads to so many avoidable deaths waiting that extra 1/10 second for it to continue turning to keep the chain running. I also like games that use a power up shot. Oh, and truly unique characters with actual strengths and weaknesses.
Dragon Blaze I adore. The dragon launch mechanic is super fun and strategic. Great pacing and music in this game. Hitting tech bonuses has a very rewarding feeling.
Zero Gunner 2 has the Psikyo trademarks, but also the Under Defeat mechanic of stages sort of "floating" horizontal, pausing, vertical and your ship can go in all directions. I think that's cool. A favorite for sure and feels like it should be easier, but i've never seen stage 6.
CAVE stuff - there's nothing I dislike, though I was underwhelmed with DDP Saidaijou AND Akai Katana. I really need to try AK again. I only played maybe a dozen rounds - I was expecting a Pro Gear / Deathsmiles feel given the horizontal nature, and there is some Pro Gear feel, but I can't put my finger on it, just left me wanting more. I really need to get at it again.
Pro Gear I simply love. I think it is a perfect game. The scoring, the flow, the music, the sprites, the boss fights, that vertical stage - the distinct color palette changes. Just a flawless game for me and my most rewarding 1CC.
Deathsmile I also love for many of the same reasons. Certainly an easier game and really only has 5 minutes of difficulty, but there are few better experiences than the orange 10K bonuses flying in. Activating your hyper on a level 3 board filled with bullets and enemies is just fantastic.
DDP DFK 1.5. This is one I love the scoring in, really great chaining, brilliant colors, great music again and wonderful boss fights. Stage 5 is super long and those spinning turrets just are something I can't master. I've had amazing runs ruined many times by those dudes.
I really enjoy the Esp series, with Esprade probably my favorite, though Galuda II looks the best. I have cleared Galuda I and II, but not RaDe.
The Mushi series is also wonderful, but I don't enjoy the scoring as much.
Muchi Muchi Pork is fun, but it feels "cluttered", which is odd in bullet hell, but it just does. It's too distracting. I say this about I game I really like. I find myself being overly critical to differentiate CAVE stuff.
Ibara...I don't get. I like it. It's hard. I remember chempop one time told me I need to use the cannon or something. I can't figure it out.
NON-CAVE/Psikyo stuff I like, well, a lot.
Top of my head go to favorites:
Under Defeat - another near perfect game
Dimahoo - uses the Ikaruga hot/cold bullet mechanic, but does it better, AND has the coolest inventory system in any shooting game ever, which is amplified by the power of your power shot, which you build up as you need. Such a cool game.
I also enjoy odd shooters, that are more than just odd for oddity's sake:
Parodius Series
Captain Tomaday
Twinkle Star Sprites
I find these games are fun for people who may not love bullet hell, want something a little funner and more quirky.