Surprised no-one has mentioned Pokemon yet... Granted, it's not much of a mini-game, but it's "fishing".

I enjoy most mini-games, and most incarnations of fishing are no exception. I appreciate that the devs took time to embed a whole other game within their main work. But I would make an exception for Skyrim's "fishing" game, which is annoying on multiple levels, not least the fact that it's now embedded into the blasted game and nothing short of high-grade, time-intensive modding can get rid of it.