The job the game has to do
A game you keep open while working is doing something unusual. It has to be interesting enough that you want to come back at lunch, and boring enough that it never asks for you in the meantime. Most games fail one half or the other, and the genre that gets both right by accident is this one, because idle games were designed around a player who is not looking.
The criteria that actually decide it are how long the game lasts, what it does with the time you were away, and how loud it is. Everything else is preference. A game that is over in three hours is a nice evening and a bad Tuesday. A game that punishes a closed tab makes you babysit it. A game with sound effects will, once, play a fanfare during a meeting.
What a background tab does to a game, technically
This is the part nobody writes down and it changes the recommendations. Browsers throttle timers in hidden tabs. A tab you have not looked at in ten minutes is not running at full speed, and after a while it may barely be running at all. So a game that counts its own ticks quietly loses time while you work, and by the afternoon it is hours behind where it thinks it should be. A game that instead reads the system clock and works out what it missed does not care how long the tab was hidden, or whether the browser was even open.
That distinction, not the genre label, is what separates games that genuinely idle from games that only look like they do.
Antimatter Dimensions
The default answer. Antimatter Dimensions calculates offline progress properly, so a closed laptop overnight is a legitimate move rather than a wasted eight hours. It is enormous, somewhere around two hundred and fifty hours to the end, which is the right size for a game that gets ten minutes of attention a day for a year.
On noise: we build it from source and serve it here, so we can be specific. It ships a small number of audio files, and none of them play on their own. One is a news-ticker clip behind a speaker icon you have to click, and the rest belong to the end credits, which carry their own mute button. In normal play it is silent.
The one real objection is what it looks like. It is a dense grid of large numbers in scientific notation, and while it does not look like a game exactly, it does not look like work either. Somebody glancing at your second monitor will not be able to tell what it is, which is either the best or worst property on this page depending on your office.
Kittens Game
Kittens Game is the one that passes the glance test outright. It is a table of resources, a row of tabs, and small text, and from two metres away it reads as an internal admin tool.
It also handles absence better than anything else here, and differently. While the tab is closed you accumulate temporal flux at a fixed rate, which you spend to fast-forward when you come back, with a separate capped offline catch-up on top of that. In other words, being away is a resource. That is close to ideal for a working day, because it turns the eight hours you were not playing into something you deliberately spend at 5pm rather than something the game silently hands you.
It is a long game, in the several-hundred-hour range, and it is genuinely fiddly. Expect to keep notes.
Trimps, with a caveat
Trimps computes offline progress on your return and has a setting for how it applies it, which is more thoughtful than most. It is deep and it stacks four separate reset layers, so it will not run out.
The caveat is that it is desktop only and it is not a two-hour-neglect game the whole way through. Early runs idle beautifully. Later, the interesting decisions are in map runs and gear timing, and those want you present.
Progress Knight, which does not fit and is worth explaining
Progress Knight has no offline progress at all. Time passes only while the tab is open and unpaused. That reads as disqualifying and mostly is, but it fails in an unusually convenient way: an open background tab is exactly the situation it needs, and it autosaves every three seconds, so a browser crash costs you nothing. It ships no audio whatsoever.
What kills it is the laptop lid. Close the machine at six and you return to precisely where you left, which is the wrong deal for an overnight game. Take it if your browser stays open for days. Skip it otherwise.
Two to avoid, specifically
A Dark Room is one of the best games in the genre and a poor choice here. It is finite, around three hours, and the parts worth having are the parts where it wants your full attention. Playing it in ninety-second fragments between emails wastes it. Play it at home in one sitting.
Cookie Clicker is the reflexive suggestion and it does not survive the criteria. Golden cookies appear on their own timer and disappear within seconds, so the optimal play is to watch for them, which is the opposite of a background tab. It also has sound on by default and a large animated cookie on screen, which is about as deniable as a slot machine.
The obvious caution
Your employer can see what your browser is doing, and whether they mind is not a technical question. Nothing here needs an install or an account, which keeps the surface small, but a personal phone is still the safer place to run a long save if the policy is strict and you would rather not test it.