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> get a job before it expires and the gov lets them stay
I never got a job here where I didn't have to provide proper documents (either my ID, or in the case of other visa-people, they've had to provide passports and documentation that they're allowed to work here, talked with those too).
It's possible, I know it's possible everywhere, but those kind of jobs are really bad and it's much harder to evade taxes here IMO.
Also I don't know if there's any naturalization law like in US, probably not and the whole healthcare system is public so almost no funny business if you don't have the ID-number that lets you use it.
I dunno, I feel it's actually much harder to be an illegal here than in US since there's nobody that will root for you like there where you have big organizations that specifically target helping those.
Maybe in the west or scandi it's different though.