NOTE: Visual novels are not listed here, but it is fine if you want to list them.

For me:

Romance of the Three Kingdoms (I & II, 1985, 1989)
- I beat both games at least 10 times, respectively.

信長の野望・全国版 (Nobunaga's Ambition, 1986)

Patton vs. Rommel (1987)
- I beat the game at least 20 times.

Quest for Glory (1989)
- I beat the game at least 10 times. At least 5 among them I chose Thief as my class.

Prince of Persia (1989)
- After beating the game, I and my friends tried to beat it within half an hour, without game saving.

Indiana Jones and the Last Crusade: The Graphic Adventure (1989)

四川省 (Shisen-Sho, 1989)

Loom (1990)

Princess Maker (1991)

Gods (1991)
- en.wikipedia.org/wiki/Gods_(video_game)

俠客英雄傳 (1991)
- You can choose to marry with different girls, one per time. There are 5 girls you can choose in this JRPG, so you have to beat it 5 times to see all the story branches.

Minesweeper (1992)

Indiana Jones and the Fate of Atlantis (1992)

仙劍奇俠傳 (Chinese Paladin, 1995)
- Many Chinese players beat the JRPG tens of times, including me.

There are many other games I have beaten, but I think I did not beat them 5 times.

There was a thread several years ago with the exact same idea.

Sure I can list a few games here but will probably not remember all.

Tetris

Minesweeper

Far Cry (at least once on highest difficulty)

Master of Orion (original) (never won on highest difficulty I suspect)

Master of Orion 2: Battle at Antares

Mount & Blade (or possibly just 4 times, not sure)

For me:

Baldur's Gate 1
Icewind Dale
Icewind Dale 2
Dragon Age: Origins
Neverwinter Nights 2
Temple of Elemental Evil
Wasteland 2

Hmmm, I sense somewhat of a trend in those games...

Arcanum
Vampire The Masquerade Bloodlines
Doom II
Fallout
FO2

That might be it. I have finished Shadowman twice. I hardly ever finish any game even once.

Hmm, games i can say for certain are:

Pool of Radiance, Dark Sun and most other D&D games like BG, Planescape Torment etc

Divine Divinity

Fallout 1,2 and New Vegas.

Heroes Might an Magic 3

Arcanum of Steamworks and Magick Obscura

Gothic 2 NOTR

Daggerfall and Morrowind

Diablo 2 LoD

System Shock 1 and 2

Deus Ex

Wizardry 8 and some of the earlier

Might and Magic VII

And a million other games i've liked.

Wolfenstein 3D
Spear of Destiny ( I like SOD better then Wolf3d)
Doom
Doom II
Final Doom: Evilution
Doom III
Quake
Quake II
CNC: Tiberian Dawn
CNC: Red Alert
CNC: Red Alert 2
CNC: Tiberium wars
Deus Ex
System shock 2

just a small sample of the games I have finished more then once.

Prince of Persia
Xenon 2
Wolfenstein 3D
Minesweeper
Civilization
Colonization
Max Payne
The Binding of Isaac
Everspace
Death Skid Marks

and so many others I can't recall...

SaGa 1, 2 (both original and DS), probably 3.

SaGa Frontier (I think).

Zelda 2. Also Zelda: A Link to the Past (though some victories were due to triggering the credits early).

Castlevania: Circle of the Moon.

Dust: An Elysian Tail.

Probably some of the early Super Mario games.

Final Fantasy 4. (Quite possibly 5, though, not living in Japan, I didn't get to play this game on its initial release.)

Final Fantasy Mystic Quest. How is that game? Is it any good?

(Also, what system is that game on?)

Doom 1 & 2
Duke Nukem 3D
Descent
Novastorm
Master of Orion 1 & 2
Diablo 1 & 2
Destruction Derby
FlatOut 2
Star Wars: Jedi Knight 2 - Jedi Outcast
Broken Sword
Red Faction
Prince of Persia: The Sands of Time
Tomb Raider
Space Ace
Wacky Wheels
Ascendancy
...

I'm sure there are some others I can't think of right now.

All the Deus Ex games except not the newest one yet.
All the Elder Scrolls games except Arena.
All the Fallout games except not 4 yet.
Crysis
Mass Effect 1 and 2 I think
Maybe Doom, Doom 2 and others from back then.

Also some Zelda and Mario games on the SNES I bet.

I would guess in the future I will get to 5 times with the Witcher, Dishonored, Prey and other similar games.

I am concentrating on single-player games with some kind of campaign or mission. Other kinds of games, like sports games, racing games, online shooters (Team Fortress 2!), strategy games with a skirmish mode etc. can be easily "beaten" dozens if not hundreds of times.

Games I'm quite sure I've beaten more times than that:

Starcraft + Brood War
Age of Empires + the Rome expansion
Age of Empires 2 + the expansions
Age of Mythology + expansions (not 100% sure about this, but I think five or more times)
Half-life + Opposing Force + Blueshift
Serious Sam: First Encounter
Serious Sam: Second Encounter
Wing Commander + expansions
Wing Commander 2 + expansions
Red Baron
Space Quest 1 VGA
Space Quest 4
Doom
Doom 2

Ummm... Is that it? There are lots of games I've beaten several times, but not sure if five or more. This is why I decided to concentrate on games with some kind of mission or campaign, as I am sure I could list dozens of neverending arcade or early video games that I've played more than five times. I mean, how do you "beat" the vanilla Tetris? Does it have some kind of final level after which you get "Congrats, you won the evil Russian king, Tetris the Mantits!"

And I guess Minesweeper is one of those "skirmish-games" where beating it can take mere minutes, and the game is designed to be played over and over again, even dozens of times in a day, usually at your workplace during the office hours of course. A bit like Solitaire.

Which reminds me, I must have finished Space Ace (the arcade) dozens of times, even in the Hard mode. I used lots of coins into it as a kid. I think the full playthrough of Space Ace, from start to finish, takes maybe 15-20 minutes. I guess there are Youtube-videos showing it how long it takes exactly.

Phew, probably more but:

Sonic the Hedgehog
Sonic 2
Sonic 3 & Knuckles
The Curse of Monkey Island

And probably a few more but these are my favorites. :-P

None. I rarely play games twice, more than that I think has never happened.

I can only think of Civ3, Civ4, and some Total War games (Rome, Medieval). AvP too, maybe.

The other games I've completed several times top at twice or three times, I think.

Edit : Oh right. Master of Magic. Forgot about that.

It is a classic JRPG that almost all Chinese players played it repeatedly during year 1995 to 2005.
Sadly, it only have Chinese and Japanese version. Fan-made translation in other languages never finished.

The game have DOS, Windows, Sega Saturn versions, and an Enhanced remade for Windows.
Note, only the 1st game (and its remakes) in the 仙劍奇俠傳 series have Overwhelmingly Positive reviews.

2nd and later games in the series have various legal and/or gameplay problems.

I think Themken means the Game Boy version.
Tetris on Game Boy do have a true end and everyone could beat it.

Some other Tetris ports or clones also have one or more endings.