I'm not a huge first-person fan, and many of the ones I do enjoy I might very well enjoy more as third-person. I don't play them a whole lot, and actually had to look through my library to pick out examples for some categories, so make of this what you will. Also, this will be a bit long, since I'm just going through and pulling up various categories, to try and work through it to give a good answer. Kind of funny that I'm not really a fan, but just spit out a bunch of games. Oh well, it's a fun thought experiment.
The first-person categories I have and/or do enjoy, though:
Your classic straight shooter, usually with multiplayer focus; the early Battlefields and CoDs, for example, Team Fortress 2. Favorites of these being the original CoD Modern Warfare (Peer2Peer servers for the win, too), and Battlefield Bad Company 2.
Roguelites; these often work pretty well as FPSs, with a relatively short but dynamic gameplay loop. A lot you can do with the basic medium. e.g. Gunfire Reborn, Roboquest, Immortal Redneck.
RPGs; this one falls in the 'I'd probably prefer third-person,' but it can sometimes work.
A bit loose, but 'thematic' shooter; Bioshock, System Shock, Deus Ex, etc. These work really well in first-person, and would be pretty hard, perhaps impossible, to do in third-person with as much feeling or atmosphere. While not my favorite category of the bunch, I would say this is near the top when it comes to utilizing the first-person perspective.
Open world, exploration/looter/etc.; e.g. the Far Cry series, Dead Island. Newer Fallouts, although these could go in the 'RPG' or 'Thematic' categories too.
Team-based survival; Left 4 Dead, Vermintide/Darktide, Deep Rock Galactic, etc. Another category that, while not my favorite, lends itself to first-person.
Doom-like; e.g. Doom. Sorry, couldn't help it. Don't play this category much, but I should. I've heard good things about some of the newer Dooms, I should give it a go. I enjoyed the older ones decade(s?) ago.
Battle royale; not really my thing (I hate the mostly obligatory very long TTK, among other things), but have had some fun with it in the past.
Survival; Subnautica. I know it gets some hate because the devs are woke retards, but I think it's an interesting and unique take, that at least does its own thing because it's in an underwater environment.
Unique; a bit of a loose category, to say the least, but things that play differently or add interesting hooks can be very interesting in first-person. Whether that's a puzzle like Portal (or any of the copycats), or time manipulation like SUPERHOT, there's plenty of room to add some unique twists.
Vehicle games; racing, dogfighting, spacedogfighting, mechdogfighting, what have you. Another category that really lends itself to the first-person perspective.
Lastly, not sure where to put this, but Guns of Icarus Online. I'd almost forgotten about that one, but played it a bunch back in the day. Another one that lends itself to first-person, and wouldn't feel as tight in third-person.
Anyway, now that I have that wall of text...which ones do I actually like best? Probably classic shooters and roguelites, and the occasional open world and stealth/tactical.
I'm not a huge first-person fan, and many of the ones I do enjoy I might very well enjoy more as third-person. I don't play them a whole lot, and actually had to look through my library to pick out examples for some categories, so make of this what you will. Also, this will be a bit long, since I'm just going through and pulling up various categories, to try and work through it to give a good answer. Kind of funny that I'm not really a fan, but just spit out a bunch of games. Oh well, it's a fun thought experiment.
The first-person categories I have and/or do enjoy, though:
Your classic straight shooter, usually with multiplayer focus; the early Battlefields and CoDs, for example, Team Fortress 2. Favorites of these being the original CoD Modern Warfare (Peer2Peer servers for the win, too), and Battlefield Bad Company 2.
Roguelites; these often work pretty well as FPSs, with a relatively short but dynamic gameplay loop. A lot you can do with the basic medium. e.g. Gunfire Reborn, Roboquest, Immortal Redneck.
RPGs; this one falls in the 'I'd probably prefer third-person,' but it can sometimes work.
A bit loose, but 'thematic' shooter; Bioshock, System Shock, Deus Ex, etc. These work really well in first-person, and would be pretty hard, perhaps impossible, to do in third-person with as much feeling or atmosphere. While not my favorite category of the bunch, I would say this is near the top when it comes to utilizing the first-person perspective.
Open world, exploration/looter/etc.; e.g. the Far Cry series, Dead Island. Newer Fallouts, although these could go in the 'RPG' or 'Thematic' categories too.
Stealth/tactical; Sniper Elite, Sniper Ghost Warrior Contracts.
Team-based survival; Left 4 Dead, Vermintide/Darktide, Deep Rock Galactic, etc. Another category that, while not my favorite, lends itself to first-person.
Doom-like; e.g. Doom. Sorry, couldn't help it. Don't play this category much, but I should. I've heard good things about some of the newer Dooms, I should give it a go. I enjoyed the older ones decade(s?) ago.
Battle royale; not really my thing (I hate the mostly obligatory very long TTK, among other things), but have had some fun with it in the past.
Survival; Subnautica. I know it gets some hate because the devs are woke retards, but I think it's an interesting and unique take, that at least does its own thing because it's in an underwater environment.
Unique; a bit of a loose category, to say the least, but things that play differently or add interesting hooks can be very interesting in first-person. Whether that's a puzzle like Portal (or any of the copycats), or time manipulation like SUPERHOT, there's plenty of room to add some unique twists.
Vehicle games; racing, dogfighting, spacedogfighting, mechdogfighting, what have you. Another category that really lends itself to the first-person perspective.
Lastly, not sure where to put this, but Guns of Icarus Online. I'd almost forgotten about that one, but played it a bunch back in the day. Another one that lends itself to first-person, and wouldn't feel as tight in third-person.
Anyway, now that I have that wall of text...which ones do I actually like best? Probably classic shooters and roguelites, and the occasional open world and stealth/tactical.