It will be hard to name only three games if we talk about pc,mobile and console.
Since, it's hard .. for me at least. I will name 3 for each device.
Vainglory - Pretty much the 3rd moba games I played after Dota and LoL. I like it cause it only will take around 15-25 mins(average) for one game to end. It's still a new game(from 2014) and so far only 22 heroes.This game also more similar to LoL than Dota because we have to buy heroes. It's only 3v3 on a single lane. It have 3 roles, Lane, Jungle and Support/Roam. I like the fact that it end it within 30 mins.
Castle Clash - The first game I played on my phone(similar to Clash of Clans). Overall, I think Clash of Clans is a bit better with their system but I like this game more. I like that we can have heroes/legends to protect our place rather than just King or Queen(in CoC). It's still a bit lacking with system like guild wars(only top 2000 guilds can war).
Avabel - RPG games for Iphone. Doesn't have solid reason why I like it. All I can say is I am a fan of RPG games. I used to play RYL, Diablo and Ragnarok(actually more but this 3 is my favs). That game is still in my Iphone but I rarely play it cause it makes the phone reallllly hot. Maybe after I get an Ipad I will continue playing it.
Brawlhalla - I know this game from this forum and I'm addicted to it. To tell you how much I have played it. I have bought Sentinel and still have around 5k on my account.
Team Fortress 2 - I played this game for around 4 months now I guess. I really enjoy the game because it's a little bit different than most shooting games like CSGO, Urban Terror and Blackshot. I still play that game because I enjoy playing it with my friends.
SAMP - SAMP is a multiplayer game(online) and its long name is San Andreas Multiplayer(SAMP). It's pretty much GTA SA online. I enjoy that game and have been playing it for 6 years now. It have a lot of servers but I enjoy roleplay servers. This game gotta be the game that I'm the most addicted too.
I only have Xbox 360 so all this Console Games is from Xbox 360.
FIFA 16 - I love football so of course I love FIFA. For console I started playing FIFA since FIFA 11 but if overall(including pc) I have been playing FIFA since FIFA 99.. I like PES too but FIFA is more realistic in my opinion.
NBA 2k16 - Pretty much the same reason like FIFA. I play basketball, so ..
I don't have the third favourite game for console.