No saving/self contained... that heavily limits it.

Hmmm... otherwise sounds more like leaning towards a solitaire, Freecell or other type of game. Or Minesweeper.

Let's see. All these will have some sort of saving, but i wouldn't consider that to mean much.
Triple Town (3 identical nearby to upgrade, ends when no more turns are possible). Beyond there being an island where you can upgrade it with gold statues or letters for whatever...
Downwell (sorta rogue-like)
Symphony (Songs are levels, and 5min or under)
OlliOlli (Skateboarding)
Race the Sun (probably...)
Super Hexagon (you may last seconds, you won't last more than 2 minutes ever)
The Pit (Rogue-like)
ToME (Rogue-like)
Tyrian2000 (Play on hardest difficulty :P )

Papers Please (when you get to the infinite/free play option, each day is maybe 10 minutes long)
Risk of Rain (Probably, longest game would be 15-20 minutes unless you're good)

Note: Rogue-likes you die fast until you learn how to play them properly and be ruthless and defensive. Think it took me 5 days to get past level 5 in ADOM. I at one time had a 20+ level firemage in ADOM.... many characer starved to death or were murdered. One even got sacrificed :P

I played ADOM for a loooong time without knowing about the command to wipe face. Which meant some caves, randomly, upon entry, would permanently make me blind. Sigh..

Says who? My record is above 2 minutes is every level and in the first one above 4 minutes.

You can also try with pinball, football (soccer) and racing games.

Would it be OK if it's 5 min per level?

'cause if so, I'd definitely recommend The Marvellous Miss Take.
Each level is just a few minutes and, IMHO, there's no real push to complete the bare-bones story.

Early experiences. I isn't impossible to do better, but the difficulty ramp after a minute completely changes. You'd have had to been playing the game for quite a while to get it that high.

I would not recommend Rogue-likes because these are typically very long games. True, you can get killed very quickly if you don't know what you are doing but I don't think that is what the OP asked for.

I can second the recommendations for

Triple Town
Downwell
Super Hexagon
Race the Sun
Symphony

and would like to add

Super Puzzle Platformer Deluxe
Spelunky

While you can't play any of the games from start to finish in 5 minutes (I'm not even sure if some of these games even have such a thing as a goal that can be reached) a typical game only lasts a few minutes and they have high replay value.

Downwell - the goal is quite literally the title of the game. Go down the well, rescue the cat.

A lot of business trips do the trick ;) especially during flights I have enough time to spare

It probably helps the soundtrack is kickass :)

There is an overarching story in Reigns that takes a few monarchs to complete.

Reventure has quite short playthroughs, but that's mostly due to quick deaths.

It's not on GOG, and I didn't like it, but the only game that comes to mind is 30 Flights of Loving.

TBH, if we're looking for 5 minute games, the Android or iOS stores would be the place to look. The people who by games here are much less likely to be looking for anything that casual. A half hour, probably or maybe even 15 minutes. But, 5 minutes is more or less at the point where I'd question the point of it.

If one were to quibble:
1) The flexiscope "timer" adjusts to your play style. So if you didn't go for full floor clearance, I'd wager you could definitely get the small down to 5 minutes to pass a floor. If you look at the speed run records, some of them are spending a lot less than that to clear 2 floors. ;)

If you go for full clear, and fight semi-cautiously, then yes, 10-15 min sounds right.

2) While outside what the OP asked for, you can save at any time, so technically you could set a timer on yourself and log out after 5 min regardless.