Let's see here. In no particular order:

Secret of Mana (1993, SNES)

Actraiser (1991, SNES)

Super Mario World (1990, SNES)

Kirby's Dream Land (1992, Game Boy)

Kirby's Adventure (1993, NES)

Super Mario World 2: Yoshi's Island (SNES, 1995)

The Little Mermaid (1991, NES)

Chip n' Dale Rescue Rangers (1990, NES)

There aren't that many I've dedicated to beating thrice.

I can give reasonable answers for a couple Tetris versions:
Nestris (Nintendo's NES Tetris): For Type-A, if you can reach level 29, I would consider the game to have been beaten. (Once you get this far, it is no longer possible to move a piece to the lower left or lower right corner, making the game unplayable at this point. Reaching level 30, and in theory 31, is possible, but it requires just the right stack and very good luck. For Type-B, simply beating level 9 high 5, or alternatively level 19 high 5 (there's a button code to start 10 levels higher) would count, and I believe it does give you an ending.
Tetris the Grand Master: One could either go with a game clear (reach level 999), or require Grand Master rank. In TGM2 and TGM3, there are other modes with other challenges (TGM2 has Death and Doubles, TGM3 has Shirase and Sakura). Note that there's no endless mode in this series.

Finishing a game five or more times, I don't think I've ever been that sad with modern gaming. Playing the same repetitious thing over and over again just doesn't have an appeal. With all the games out there, I'll probably play one maybe two or three times over in ten years. The only one I can think of that got that far is old Megadrive games like Golden Axe. The kind of game you can finish in an hour.

There are some games that I like to return to and that draw me in enough to make me finish them once I start, even though I know them already:

Ultima 7 - more than 10 times, I think.

5 or more:
Max Payne 2
Mafia
Monkey Island
Loom
Diablo
Diablo 2
Raptor
Super Mario (Gameboy)
Gargoyle's Quest (Gameboy)

Yeah I am sure there are all kinds of Tetris versions, clones, spin-offs, sequels etc. there. The Tetris games I recall playing didn't really seem to have any real ending, as far as I recall. I think I played the Amiga version of Tetris, and later some generic Windows version of Tetris.

At least the Windows versions seemed to just be an endless supply of bricks falling, with increasing speed the more you progressed. I presume it had no ending, just trying to get a high score. A bit like playing a pinball machine.

Not sure what is considered as the "original" Tetris, is it one of those home versions for Commodore 64/Amiga, or the arcade version, or what. I think the arcade version(s) was a port from the home versions?

Real™ Original Tetris:
https://www.youtube.com/watch?v=O0gAgQQHFcQ

The winner for me is Return to Castle Wolfenstein - definitely at least 10+ times

Then:
Doom I, II, TNT, Plutonia and many other full IWADs
Duke Nukem 3D and some fun with EDuke32
Blood + expansions
Red Alert 2
Overlord I
International Rally Championship (and a lot of splitscreen with my cousin)
LotR BfME 1
LotR BfME 2 + expansion and countless War of the Ring in 2
Serious Sam TFE
Serious Sam TSE
Serious Sam BFE
Diablo II
Diablo 1 - not sure if 4 or 5 times
Unreal
Unreal Tournament (Many times, including Godlike difficulty)
Unreal Tournament 2003 (Many times, including Godlike difficulty)
Unreal Tournament 2004 (Many times, including Godlike difficulty)
Unreal Tournament 3 (hardest difficulty)
Star Wars Jedi Knight: Jedi Academy
Grim Dawn (at least first difficulty, one character on all difficulties)
Titan Quest (at least first difficulty, one character on all difficulties)
Asterix The Gallic War
POD
Carmageddon 2
Die By The Sword + LFL
Rayman 2 (still by far the best one)


Not sure about others, could be some more I can't remember right now or some I played 3-4 times.

"real" finishes* 5+ times for me is a very short list, I don't have infinite time and lots of things to play.

Baldurs Gate 2 (PC) + expansion 12+ times ;p
Dragons Dogma: Dark Arisen (PC) so much variance in classes and how they play
Factorio (PC) ie. launched a rocket, at least 7 times on new maps before mods

And yeah, Diablo,Unreal Ut2k4 etc. but those don't count as real finished games imo. UT2k4 was only a competition skirmish kind of thing, and Diablo is infinite and I played each Diablo with each class through.

I am one of these people who start a TES game and never finish the main story as I get lost in modding. Same for many other games.

Last but not least. Total War: Warhammer 1 and 2. Yes, I beat the main campaign goal (long) for each faction. And there's a lot of factions.... ^^ Same for any other Total War game.

And nearly forgot. Dishonored 1 and Half-Life 1 -> speedrunning

Well, and every Anno game ever made.

If we go by that then I guess I can count them too ;) My biological memory has deleted most of these things heh

I don't think I've ever beaten a game that often, not even games like Wacky Wheels / Skunny Kart or Bubble Bobble with a friend. 2-3 would already be a *very* rare occasion for me (maybe Monkey Island? Dyna Blaster?).

Since it is part of the mechanics: Tower of Guns

edit - may expand. Each successful run and achievements unlocks new weapons and perks. To get 100% complete, you need to beat the final "secret" boss with each weapon and perk ('star' them). Since there are 10 different weapons and 10 traits, to get close to 100% you need to beat the game at least 10 times.

None :(

Need for Speed Underground.

Stopped counting how many times I've beaten it a long time.
Don't think there was ever a time I didn't have it instaled either.

Alien: Isolation
Diablo 1, 2 and 3
Final Fantasy 7, 8 and 10
Grandia 2
GTA 3 and Vice City
Hunie Pop
Master of Orion 2
Star Wars: KotOR 1 and 2
Super Mario Bros. 1 and 3
Sword of the Stars: The Pit
Torchlight 1 and 2
Wing Commander 3 and 4
XIII
Zelda