Touhou comes to mind for me. Maybe it doesn't really qualify since I wasn't that interested in shoot-em-ups before trying it, in fact Flying Tigers is the only one I can vaguely remember. But my love for Touhou is definitely not despite the bullet hell mechanics, where you try to avoid myriads of colorful bullets forming flowery patterns while struggling to deal some damage yourself. I guess, it just felt like a shoot-em-up on steroids, an extreme expression of the mechanics I found in games before it, that somehow managed to move away from the military/sci-fi image my mind had linked them with. Like in anime when characters unlock new forms they look more brutish and beast-like, but the ultimate form is very small and humanoid by comparison, the kaleidoscopic movement of colors, the evoking music, the manga-esque character design make it feel like something different.
EDIT:
Oh, and another genre -- space sims. Well, actually it's two games, but for different reasons. First one is Battlecruiser Millenium. Yes, it's not very rewarding and I wouldn't say it even good or something I'm looking forward to playing again, but the freedom it allowed spoiled me. You can start as a commander on a giant starship outside a space station orbiting Earth. You can engage pirates in a space battle and send out interceptor squadrons to aid you. You can shoot rockets from the main ship or get into one of the interceptors yourself and dogfight while giving orders to the rest of the squad as well as the starship. You can suit up for EVA and shoot the fly-by craft with your rifle while walking on the ship's exterior or jet-packing 'above' it. You can jump to other systems or send out smaller crafts or even probes to explore them. You can enter the atmosphere of planets and their moons. You can land on them or just fly around. If you landed, you can deploy ATVs and drive around the surface. You can deploy mining drones and take a walk outside on your own two with your trusty guns while it is commencing. You can have your security personnel escort you or have them do the walking while you stay in the ship. You can have them attack enemy bases scattered around planets or just nuke them from orbit.
Unfortunately, for me it's not as fun as it sounds. I really wish someone other than Derek Smart had made it. I've seen other sims adding more depth by allowing to visit the planets, but nothing this deep so far.
The second game or game series is Space Rangers, mostly Space Rangers 2. It's a bit different in the sense that I think it's actually a very good and enjoyable game and that its thing could've possibly worked in other settings outside space sims, while still, in my opinion improving it as a space sim. It's what I found lacking in the previous game -- depth. You play as a space ranger one of the chosen who must aid the military forces in repelling the impending invasion of an alien hive mind(SR1) or space SkyNet(SR2). The allied forces themselves are composed of five species from which you can freely choose one as your own. The species have different physiology and culture, leading to preferences in the way of profession -- trading, pirating, pirate-fighting, etc; supply, demand and restrictions for goods on their planets; tech levels of available equipment; as well as interracial relations ranging from prejudice to friendliness which influences the way other rangers and planetary governments treat you and other rangers. Yes, in this game you're not the center of the universe. Other rangers can overshadow you in fights against enemy threat, just like you, they can hunt pirates and be pirates, they can go to jail. The game's world lives and breathes without you. Different events happen on the planets leading to trading opportunities, the military organize system re-capture actions, etc. That's just a gist, there's a lot more stuff in this game.
I've seen some sims playing with either of the two features, but never to the same extent.
