What the filter is actually blocking
Your school is not blocking idle games. Filtering products do not know what an idle game is. They work on domains and on categories, and somebody in IT ticked a box called Games, which switched off every domain the vendor has filed under that heading, all at once. That is why a well-known clicker and a random portal hosting the same clicker can vanish on the same afternoon, and why one of them sometimes reappears later while the other never does. Nobody made a decision about either game.
Four properties get a page cut off in practice. The domain being categorised as a games portal is the big one, and it is the reason the popular mirror sites keep dying: they are all in the same category list, so they get blocked in batches rather than one at a time. User-to-user chat is the second, because chat is a separate category that most schools block harder than games. A login is the third, and it is less about blocking than about the school not wanting accounts it cannot see. The fourth is the quiet one: a page can load fine and still fail, because the game's script comes from a CDN that is blocked, or an analytics call hangs the loader. You get a white rectangle and no error, and it looks like the game is broken.
The shape that keeps working
A game with a decent chance of loading on a locked-down network is one static HTML page with its scripts sitting on the same domain, no account, no chat, and a save kept in your own browser. That is not a trick. It is just a smaller target: one domain to be categorised instead of four, and nothing on the page that trips a second category.
This is also why the genre is a good fit for the situation in the first place. Most of the respected incrementals are, technically, one page of JavaScript and a save string. They were built that way for reasons that had nothing to do with school networks, and it turns out to be the shape that survives.
Two we host ourselves
We build Progress Knight and Antimatter Dimensions from their own source and serve them from this domain, with every third-party request removed. No analytics inside the game, no CDN fetch, no font server, no sign-in. Both are one page that loads and then stops talking to anything.
Be clear about what that does and does not buy you. If this site is blocked, those games are blocked, and stripping the third-party calls does nothing about it. What it does buy is that they cannot half-load. There is no second domain that can be filtered out from under them, so either the page opens and the game works, or the page does not open at all.
Progress Knight is the better one to actually start in a free period. It is a medieval career ladder made of progress bars: you beg, you get a job, you age, you die at seventy, you rebirth with permanent multipliers. It is mouse only, it ships no audio at all, and it autosaves every three seconds. It also has no offline progress, so it only runs while the tab is open, which matters less when you are sitting at the machine anyway.
Antimatter Dimensions is the opposite proposition. It is roughly two hundred and fifty hours deep, it does calculate offline progress, and it will still be giving you new systems months later. It also assumes a wide window and a lot of visible numbers, so a cramped lab monitor is not doing it any favours. Start it if you want something to keep, not something to fill twenty minutes.
The ones we can only link to
A Dark Room is the best short recommendation here and it lives on its author's own domain, which is a single-purpose site rather than a games portal. It takes most people two to four hours, it ends properly, and it has a sound toggle you should find before you click anything. Trimps and Kittens Game are both excellent and both far longer, and both depend entirely on domains we do not control. If they load for you, take them. If they do not, there is nothing to be done from this end.
When something is blocked
We do not publish proxy lists, mirror domains, or instructions for getting around a filter, and we are not going to. On a school-managed device the filter is a security control, the machine is logged, and the realistic outcome of poking at it is not a game, it is a conversation with someone about your account. The cost is wildly out of proportion to the reward.
The practical answer is almost always to play something that does not need the blocked domain, which is the whole point of the list above. The other answer, which sounds naive and is not, is to ask. Filter categories are broad and schools do sometimes unblock a specific site when a student asks for a specific reason. It costs one email.
One thing worth doing on a shared machine
Browser saves do not survive a wiped profile, and you will not necessarily get the same computer next lesson. Every game named here can export its save as a string you can paste back in somewhere else. Do that before you have forty hours in it, not after. It is the single most common way people lose a long run, and it has nothing to do with filtering.