On this site what game is most considered a good game that your addicted too that I might enjoy?

Well, what kind of interests you have? A game that's super addicting to one person might be really boring to another.

But for me, I'm addicted to the Touhou series right now. If you don't know what it is, it's basically a series of shoot em ups, but they involve a lot more dodging than shooting. If you want to know what I mean, look them up on Youtube. x3

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Well I think RGP's are kind of addictive, I mostly play RPG games so you might be able to try some out.

As someone said, it depends of the kind of games you like. 

Right now, I'm addicted to Mario's Picross 2, it's a cool puzzle game, and sometimes it's really tough

Sorry I did not give out any help.
I should have said what type of games I do like.
I like RPG games and MMOs MMORPGS.
Example as of Final Fantasy

I'll just list all the games I really like, and say why, and say what genre the game is in. The genre might just be one I made up, though. If it's an MMO that has had the plug pulled on it, I will not list it... I loved Earth Eternal, but it died due to the company that was handling it's second attempt at breathing kinda being forced to work on five too many other games at once. Small team + that = Hi sinking ship. That aside... List time!

The Main series Pokemon games(RPG)
Monster collecting. Somehow hard to get bored of gameplay for gluttons towards level grinding. I grew up on this game series, and only skipped the recent games because I'm so busy with 15 other games. Maybe 30 other games. I buy so many games at a time that... Both my shelve space and Pokemon playing is lost.

Okami for the PS2 and Wii(Action-Adventure)
I heard the Wii version is clunky, so just get the PS2 version. Basically a game that is often compared to Zelda games, but I'd say it's better. I have never played a 3D Zelda title in my life, though.

Psychonauts for PC, original Xbox, and PS2(3D Platformer)
I actually got this game because it looked interesting when I was about 12 or 10. It's always been a favorite of mine. The only bad part of this game to me is the Meat Circus level. Warning: It is best to bother with that last level once only. It is a level that probably might inflict imaginary psychical pain. This game is an extremely well made platformer that is more of a cult classic, but it's a great game. Great humor and great gameplay. It's also a blast to jump into minds.

Doom and Wolfenstein 3D for... Alot of systems including the PC. Forget the SNES version of Wolfenstein 3D exists(First-Person-Shooter)
Both are awesome games from the 90's that are extremely fun. Can't go wrong with either game if you want a good shooter. Besides, id Software are masters at making shooters. Rage is another great FPS to delve into by id, by the way. It is all the RAGE! To those who like it.

Kingdom Hearts for the PS2(Action RPG)
I like the rest of the Kingdom Hearts games except for the ones I haven't played and that remake of Chain of Memories. Only reason I haven't played much of the series after 358/2 Days(that and Chain of Memories are high recommended. Just play the GBA version of Chain of Memories. The PS2 version kinda loses the charm of the 32-bit GBA game) just because once the PSP game was released, I had no way of playing that, and sorta found other games. Yeah. Guess the only thing keeping me from losing interest was having the systems to play the games on. The first game is really good.

Silent Hill 2 and 3 for the PS2 and PC(Survival Horror)
Both are amazing games. Only thing holding the first Silent Hill back is the controls. If you like survival horror, you probably have already played Silent Hill 2. If you haven't played SH3, check it out. It's one of the most creepy, unsettling, and horrific yet awesome gaming experiences I have had.

Rogue Galaxy for the PS2(Action RPG)
Great battle system. Fun story, yet will seem bland for seasoned RPG players. It's also got a talking bulldog guy who served the military.

Soul Reaver 2 for the PS2(I... Guess action or something)
The first Soul Reaver is great, but this one just feels less confusing. Maybe I just suck at figuring out how to read PSN virtual manuals, or am too lazy to. Otherwise, the first one is considered a great gothic whatever it is game. It also has a zombie-like soul reaver using soul eating vampire voiced by an old man. And the game is fun. And the old man is good at sounding young!

Max Payne for the PC and PS2(Shooting Frank Miller comic thing genre whatever it is... I'll just call it a game with guns and gangsters getting shot)
It's a well-made game that is quite fun. And has an awesome thing to say in real life. MAX TO THE PAYNE!

Spider-Man 2 for the PS2, original Xbox, and Gamecube(Free-Roam)
I think the idea of playing as Spidey in a free roam game is supposed to sound awesome. I should ask some scientists who haven't been bitten by radioactive spiders.

Dark Cloud for the PS2(Action RPG and Dungeon crawler)
I actually have trouble enjoying this game. It is pretty cool and interesting, but the weapons system isn't fun to me. Might be more fun for you. Oh, and it has town building and the first town has a guy you wish you didn't get to see bathing.

Now just because I'm sleep deprived, and my arms are now hurting... You get a less fun list of things that are awesome games.

Demon's Souls for the PS3(really hard action RPG)
The first three Ratchet and Clank games for the PS2(Shooter and Platformer)
Ratchet and Clank Future A Crack in Time and Ratchet and Clank Future Tools of Destruction for the PS3(same genre as the first three games in the series)
The first three Jak and Daxter games for PS2(first one is a 3D platformer while the other two are more or less like GTA with elf things that are in a more story driven game)
Parasite Eve for PS1(Horror RPG)
Suikoden for PS1(RPG)
ATV Off Road Fury for PS2(Racing and driving)
Tetris for way too any systems to count(Puzzle)
Bejeweled 3 for PC and other consoles(Puzzle)
The Lion King for SNES, Gameboy, and Genesis(Platformer)
Saint's Row 2 for PC, 360, and PS3(Sandbox)
Phantasy Star 1, 2, and 4 with the first being on the Master system and the other two mentioned being on the Genesis(RPG)
The first three Spyro games for the PS1(Platformer)
Fallout 2 for PC(RPG)
The Elder Scrolls V: Skyrim, and also Oblivion for PC, PS3, and 360(Action RPG)
Resident Evil 4 for the Gamecube and PS2(Action Horror)
Dead Space for PS3 and 360(Action-Horror)
Devil May Cry 1, 2, 3, and 4 with the first three being for PS2 and 4 being for the 360 and PS3(Action)
Darksiders for 360 and PS3(Action)
Final Fantasy V for SNES, Playstation, and GBA(RPG)
Legend of Zelda Minish Cap(Action Adventure)
Sonic Rush and Sonic Rush Adventure for DS(Platformer)
Suikoden III for PS2(RPG)
Fallout 3 and New Vegas for PC, 360, and PS3(Action RPG FPS)
Ultima VII for PC(RPG)
Tekken 5 for PS2 and Tekken 6 for 360 and PS3(Fighting)
Golden Axe for the Genesis(Beat 'em up)
Picross 3D for DS(Puzzle)
System Shock for PC(sorta RPG and mainly horror shooter)
Bioshock 1 and 2 for PC, 360, and PS3(FPS)
Borderlands for PC, 360, and PS3(FPS action RPG)
The Sly Cooper series for the PS2 and its HD collection on the PS3(stealth-platformer)
Metal Slug 7 for DS(Arcade style side scrolling shooter)
The World Ends With You for DS(Action RPG)
Luigi's Mansion for Gamecube(parody Survival horror)

If you need any other suggestions, tell me. I probably know of something. Only game genre I believe I have a limited interest in is racing games, so yeah. I mainly play shooters, action games, free roam and open world games, RPGs, and puzzle games. Oh, and click and point adventure games. I love those old little things.