My forte would be Shmups (particularly of the danmaku, or bullet hell variety), speedrunning, and Beatmania IIDX.
Self-improvement is a huge part of this. My first "skill game", if you can call it that, was Guitar Hero. My friend brought the game over and played on the hardest difficulty, and I thought it was impossible. He left it over for the weekend, and I probably played 20 hours that weekend. When he came back, I was playing the same difficulty he was, and we were able to co-op some of the hardest songs in the game. It made me happy, and I could see the joy in him that someone actually gave the game a chance, which for some reason made me even more happy.
Once I discovered shmups, it was pretty much over.

Its really hard to explain why playing these games this way is fun, why losing over and over is enjoyable, and particularly that we aren't trying to show off. This isn't a way to wave an e-peen, this is a genuinely fun way to approach... well, anything!
I think the aspect I enjoy the most is segmented learning. Its something I do on the guitar, to improve my Rubik's Cube speedsolving, and to learn and perform well in shmups and speedruns. Take a small, 15 second segment of something that just seems impossible, practice it, understand it, and that feeling when you go "holy shit I think I get it.... I DID IT" is unbelievable.
Stringing those learned segments together, and creating a run, whether it be in a speedsolve, music piece, speedrun, or score attempt, is just... I don't know how to explain it, but its so personal. Its such an extension of my capabilities, and I can always clearly see where to move forward, what to learn next, while being able to constantly appreciate what I have already learned.
There are a few reasons why Danmaku shooters in particular are my favorite when it comes to Skilled Gaming, but what really makes them appealing to me is that the developers made these games with the intent of them being played like this. When I speedrun Sonic, I know I am breaking the game, I'm not playing it as intended, and I could very well run out of things to learn, hit a "ceiling" of sorts. With Danmaku shooters, I know the game is going to play well throughout, no matter how hard I push it, and in fact the games tend to get better the harder I push into them. I know that the ceiling for these games is ridiculously high, and that means that I can learn more, and make more use of what I have already learned. It feels like a never-ending battle, and that is really great to me.