Games That Were So Good That You Played Them Again and Again.

Fallout 1 + 2 + Tactics
Jagged Alliance 1 + 2 (JA2 is easily the game i have finished the most)
Mount and Blade: Warband
Kotor 1 + 2
Mass Effect 1 + 2 (i will try to finish ME3 a second time before i start with Andromeda)
Vampire: The Masquerade - Bloodlines
Neverwinter Nights 1 + 2
Baldur's Gate 2
GTA Vice City, San Andreas, IV
Saints Row 2 + 3
Arcanum
Doom 1 + 2
Duke Nukem 3d

Baldur's Gate 1&2, Fallout 2, Neverwinter Nights, Diablo 2
Great memories

Alan Wake, Quantum Break, Fallout 3, Dark Souls, Dark Souls 2, Doom, Gears of War 2, Pokémon Emerald

Super Mario Bros.
That game is just perfect in every way.

Wizardry 7
It's an old-school RPG, so it has all the issues that old-school RPGs had. but it has something I've never seen in another game, which is that there are NPCs competing with you, like it's the real world. Essentially (and blanking out the story), you're on a hunt for a number of maps. However, some NPCs are after them as well, for various reasons. If it takes you too long to get to a map, you may find it's already gone. You then need to find out which NPC has the map, and hunt him/her down. of course, the NPC is moving around, so if you know the NPC is in town X and you go there, the NPC may have already made his/her way to town Y. Or, you get there, but the NPC has already lost the map to a different NPC. On top of that, the NPCs may also be looking for you, to get the map away from you. Not only by fighting, but maybe by pickpocketing you.
Couple that with a great system for managing stats (stats improve by use as well as by leveling up), over a dozen races and a dozen classes, and an interesting, if annoying, magic system, and you've got a damn fine RPG

The original Unreal. I'm actually replaying it currently on Steam, for like the 13th time xD

I would say The Elder Scrolls series :3 Especially Oblivion and Skyrim \o/
The other one I would play over and over is Mass Effect.

The Curse of Monkey Island and Broken Sword I & II.

Dunno why but I only can play a game through just twice. And that includes Tomb Raider 2013, Witcher 3 and a few more that I don't even remember

Dragon Age Origins. I cant even remember how many times Ive played that. Id pick the "same" three characters over and over between the races, just make different choices, then restart on a different race and class.

Oh and Need for Speed Most Wanted. Loved the cop chases. But for that game, my mom has me beat. We both got it at release roughly, but to this day shes still playing it every weekend, on a PS2 that I refurbed and gave her when I moved lol

Also the Fallout series, especially 3. 3 might look a bit bland, but Ive been to most of those places in real life, so its a little extra special for me

Max Payne 3 - 9 times
Arkham City - 6 or 7 times
Call of Duty 4 - at least 10 times

disciples 2, kingdom, fallout 2, Streets of rage/SoR2, NOLF 2, BG&E, Space rangers 2, FEAR, FOOTBALL MANAGER, Civilisation

Final Fantasy VI, VII, IX, X (8 sucked simply because of drawing magic out of enemies)
Just Cause 2 (don't own 3 so can't comment)
Fallout 3 (never got to play NV or 4)
Saint's Row 3 (avoid 2 because bugs, and 4 because it sucks, also bugs)
TES 3, 4, and 5: Morrowind, Oblivion, Skyrim (the games are good, mods make them better)
Super Smash Bros. Melee (Brawl sucked compared to Melee)
Diablo 2: LoD (and the massive amount of mods)

Severance and Clive barker jericho I have finished them multiple times

S.T.A.L.K.E.R. game series
Dark Souls III
Dragon Age Origins
UFO: Enemy Unknown (AKA X-COM: UFO Defense) and my guilty pleasure X-COM: Apocalypse
Dungeon Keeper and the sequel (I am still waiting for the third one :<)
Hitman: Blood Money

Terraria
Chrono Trigger
Earthbound
Mega Man X/X2/X3

Borderlands (1,2,pre-sequel), Final Fantasy series, Counter-Strike all versions, Half-Life 1,2, L4D 1,2, Undertale, To The Moon, Bioshock, that's all for now, I'll add more later

Rock Band 3
Skyrim

games i replayed... New Vegas becuase there are different ways to tacle some missions and different concequences, Morrowind cause theres just so much interesting stuff to see.

Half Life 1
Mafia 1
GTA: Vice City
Dark Souls
Dark Souls 2
Dark Souls 3

Can't think of more atm

Europa Universalis-series
Hearts of Iron-series
Victoria-series
Crusader Kings-series
Super Metroid
Majesty: The Fantasy Kingdom Sim
Civilization-series (not counting 3)
Probotector

Mount & Blade Warband and Napoleonic Wars expansion

Gothic II