Ultima 2 has time travel. (Note, however, that many regard this entry as among the worst in the series.)

Ultima 1 also does, but not until the end.

Braid

the page description reads:
Braid is a puzzle-platformer, drawn in a painterly style, where you can manipulate the flow of time in strange and unusual ways. From a house in the city, journey to a series of worlds and solve puzzles to rescue an abducted princess. In each world, you have a different power to affect the way time behaves, and it is time's strangeness that creates the puzzles. The time behaviors include: the ability to rewind, objects that are immune to being rewound, time that is tied to space, parallel realities, time dilation, and perhaps more.

https://www.gog.com/en/game/braid

I think everyone can agree that no game in the franchise is worse than 9.

Also regarding time travel which hasn't been mentioned yet are those best gameplay levels from Dishonored 2 and Titanfall 2 respectively.
Granted, Dishonored 2 has another best level, but that one doesn't feature time travel.

Future Wars, point and click adventure where you travel to different time periods. Not on GOG but definitely should be. Although if they did, they'd no doubt get the garbage DOS version instead of the Amiga version...it has far better sound/music, which is a big part of the atmosphere.

It's a board game. It has a digital implementation on one of the board game sites, I believe. I haven't played the game and doubt I would like it, and it's a HEAVY game, but people who like it REALLY like it. It's listed as allowing for just 1 player/has a solo mode.

Anachrony.

It's a time travel game where in you can borrow resources from your future self. And hope you pay them off in time!

---

Another time travel board game that works well and is fun is Time Stories. It's listed as 2 player, minimum but could probably be soloed. I played through the whole first arc. It's equivalent to a "point and click adventure game" with a strong meta story and decent puzzles.

Fair warning: the "meta game" is completely and totally fucked by the end. Each of the individual titles stands out well with pros and cons, and the meta-story is good for a while until they totally pulled the awful ending out of nowhere/their asses -- which REALLY soured a lot of people on it. I don't think it has a digital implementation, or at least an official one. Since it has no replay value, used copies may be able to be found cheaply, and I believe it was localized well in various languages.

---

Time Loader on GOG is a fun light puzzle game with time travel as its caveat, including a few "changes in one time affect the future" parts you need to toggle to get all the endings. The whole game is a "play in one weekend" kind of game. It has a demo.

Your edit made me laugh.

When did HOMM2 have time travel? Surely just the lore of one of the characters and not a gameplay mechanic.

Fun fact: There's been time travel in Magic: The Gathering.

Time Walk allows the player to take another turn. (Pre-release, the card said "target player loses next turn", which could be interpreted as losing the game, so that had to be changed.) Note that this card is one of the Power Nine, and therefore expensive and banned in most formats, but the effect would appear on other cards.

There's also Timetwister, which basically restarts the game. It, again, is banned in most formats.

Rocket-Powered Turbo Slug, a card from one of the un-sets (basically a joke set that isn't tournament legal), has super haste, allowing it to attack the turn before you cast it.

I could also mention that some other games have time-altering magic, like Time Stop in Baldur's Gate 2, or Quick in Fell Seal and Crystal Project. In both cases, these spells give the caster or target extra turns.

Life Is Strange

Few years ago I played it on ps4 and was very surprised - game was strange but good, and I liked the story.

About time travel in this game:
Actually, you can rewind time throughout the game here, but only for a few seconds. However, later in the game you can go further into the past and change some parts of the storyline.

Cthulhu Saves Christmas has a weird form of time travel in that it continues to be Christmas Eve day after day. You'll have Cthulhu go to bed for the first time, and the next day is Christmas Eve the 2nd.... and so on. The whole plot of the game is that Christmas Day isn't arriving because it needs to be saved.

The thing is that even though it is technically a time loop, it's one where everyone is aware of it plus you keep levels, equipment, skills, and don't need to refight bosses.

-----

Day of the Tentacle is the sequel to Maniac Mansion which involves three humans working in the past, present and future to stop Purple Tentacle from taking over the world.

-----

Neverwinter Nights has time travel in chapter three of the Wailing Death campaign during the Creator Race Ruins, which is one of the better parts of this campaign along with Charwood (also in chapter 3.) The time travel use is okay and isn't the central focus of this campaign.

The user created module Prophet (which starts with "Prophet - Prologue - It Cannot Be Denied") and can be downloaded free off The Neverwinter Vault handles time travel a lot better than anything else I mentioned. There are thee chapters that follow the prologue. A lot of details are outright spoilers, but the only thing that occurs very early is that a village gets slaughtered by human knights in red armor from over 200 years ago.


-----

The Sexy Brutale also has the time loop style time travel. I do own this one but haven't played it yet. Gameplay involves exploring the mansion, hiding, and learning the circumstances of everyone else's death so that you can prevent it. However my understanding of the game is that once you've prevented one death, you won't have to redo it once the day loops around. It remains prevented.

-----

The Silent Age - I don't know much about this one except that I know it involves time travel.

While not on gog. there's the freeware RPGMaker game Rxcovery which sort of has a time loop. Thing is, you have 30 minutes to get funds to pay your hospital bill, but when time runs out, there's a New Game + feature that lets you keep plenty of things, like stats, equipment, and even quest items.

Costume Quest 2 has time travel

Thanks! I'll have to check it out, especially since game loops are only about thirty minutes. I'm really hesitant to download anything new that would be piling hundreds of game hours onto by backlog.

Back to the Future (rest in peace, but I still got my installers)

Altered Destiny

The Infectious Madness of Doctor Dekker
Well, at least one guy claims he time travels and another one has an extra hour each day in which everything but him stands still.

Echo
Shot, die, repeat!
Learning enemies who adapt to your playstyle up to a certain level

Legacy Of Kain

Minit
Just one minute to save the world ... sort of

Randal's Monday
Time traveling on LSD

Terminator Resistance

(hopoefully soon) Randal's Tuesday
Check out the quotes in the kickstarter trailer :)

And I'd love to see Death Loop here, also with good timestream altering mechanics.

The most complex game known to me when it comes to time travel however is one not on GOG: Timeloop Hunter, an adult VN style game with strong adventure game aspect, but with so many variations, that it can be a pain to find all possible solutions (which sometimes you have to. Here's a simple one: Let someone get run over by the bus so you can get information about that person which you need for a conversation which you need when saving that person).

Check out Achron. It's an RTS. Kind of ugly, and the pathing can be a little frustrating, but the engine it uses is really neat. It handles alternate timelines in the background and is fairly complex. Don't expect a plain RTS that only has a time travel theme. Look for gameplay video at Youtube, first.