Somehow I missed this year's thread, so here's what all I've completed so far...
The Howler - This was a tough little game that had a crazy difficulty spike about halfway through, at least for me it did. There were like two levels halfway through that I struggled with and in fact I stopped playing at one point. But I came back, finished those tough levels, and rolled through the rest of the game.
Turok: Remastered - Ah, Turok! The original game was one of the main reasons I bought an N64 back in the day. I never actually finished it since I got into Goldeneye, Banjo-Kazooie, and various wrestling games more. I have a copy of the PC version on disc and surprisingly, it works fine on Windows 7. I tried playing it a year or so ago and the FOV was really disorienting. Luckily, that has been fixed up, along with many other things and this is the definitive version to play, in my opinion! I loved this game and very glad I was able to knock it off my list.
Kung Fu - Not much to say here, I've played this game so many times.
Mushihemesama - Very cool shoot 'em up that I need to go back and play some more!
Shaq Fu - I played this with some friends and yeah, it's pretty bad. It wasn't as bad as I expected it to be, but it wasn't good. The controls are just bad.
Bad Dudes - I picked up the Data East collection on the Wii and played through this for the first time. It's a good thing I had unlimited continues.
Magical Drop III - These games are pretty cool and this one was kinda tough.
Heavy Barrel - Eh, I didn't like this one at all. This is another Data East game I played on the Wii, so maybe the Classic Controller just wasn't a good fit for it.
The Ultimate Doom - I've beaten this one quite a few times and it's always a blast. I'm currently working on Doom II, which I have NOT finished before.
Disruptor - A promising, but ultimately disappointing early FPS on the PS1. It was a neat game to play since it's essentially the precursor to Insomniac's Ratchet and Clank series and does indeed share some elements with that great series. But the erratic difficulty and limited controls on the PS1 controller hold it back.
Giga Wing - What can I say, it's a fantastic shoot 'em up!
Beyond Good and Evil HD - This was my first time playing the HD version, but it's exactly the same game. I've beaten this game several times and still really enjoy it. Bring on the sequel!
DmC: Devil May Cry - I tried this one on PC before, but found the controls to be a little overwhelming. I picked up the Definitive Edition on PS4 and absolutely loved it! I'm amazed at how many people did not like this game.
Astal - This is an amazingly beautiful game on the Saturn. It's been on my list for a LONG time and it was nice to finally play through it. It's a tough one too.
Donkey Kong - Always good to play this one :D
Plantera - A pretty standard clicker game that was quite addictive. It didn't take long to see everything and get all of the achievements, but I was hooked.
Donkey Kong Jr. - Always good to play this one :D
Shovel Knight - A tough, but fair old school game. No way I beat this one without checkpoints.
Saviors - I played this one at random and it was alright.
Samorost 2 - A short adventure-ish game that was really neat. I had been wanting to play it for a while.
Teenage Mutant Ninja Turtles (1989 Arcade) - I used to play this on the NES all the time and it's been MANY years since I last played it.
Intergalactic Bubbles - This is a Puzzle Bobble clone type game that's not particularly great, but I managed to spend 14 hours completing all levels and getting all achievements. So it did something right :P
Brave Fencer Musashi - I finally finished this one! I really liked it, but the last level / world kinda sucked since you lose some of your abilities.
Golden Axe: The Duel - Another beautiful 2D Saturn game here.
Super Mario World - It had been a long time since I last played through this one, but it's one of my favorite Super Mario games.
The Guardian Legend - Just my yearly playthrough :)
Super Metroid - Prior to this one, I had only actually finished two Metroid games: Metroid II and Metroid Prime. I had heard this was one of the best ones and I tend to agree. It's a very good game, though I must admit I've been spoiled by modern games and their generous save systems and map interfaces.
Mega Man 3 - Ugh, this one was really tough. I thought Mega Man 2 was a lot more fair than the first one, but this one seemed to ramp up the difficulty again. Lots of cheap sections in the game, but it was still enjoyable. I'm playing these on Steam, so I'm abusing save states haha.
RayStorm - I picked up Taito Legends 2 on PS2 and decided to give this one a whirl. I have the PS1 version, but it's a tough game and without unlimited continues, there's no way I'm completing it. It's an okay game, but I prefer the prequel.
In all, I've completed 29 games so far this year, but many of them were super short. The shortest game was Donkey Kong on NES at 3 minutes (enough to beat the three levels). The longest was only 16 hours and that was for Brave Fencer Musashi.
Here's my full list with some more info, including completion times.