I feel bad getting into it here because I feel like I'm already implicitly derailing so I'll give you a few examples and if you want to really get mincing I'll be happy to over PMs. I should preemtively reiterate that my original question was only for competence, ie: not playing like DSP, iJustine et al. rather than 24/7 uber-skilled high-level play. The most important distinction to make is between SP and MP, since they require different skillsets. On the MP side the single best game is probably CPMA for Quake 3, followed by in no particular order Brood War, SF II, Skullgirls, CS 1.6, Frozen Synapse, and Killing Floor.
On the SP side: Defrag for Quake 3, Doom, Hotline Miami 2, Jagged Alliance 2, IVAN, Super Meat Boy, Zone of Alienation, Rogue Spear, UT99.