Nuclear launch codes. Locations of strategic assets (eg, nuclear missiles). Identities of people in Witness Protection. Closed court records regarding children. Personal information collected as part of tax collection (eg, income). Identities of foreign informants.
Here's the difference: the government is quite open about doing all those things. We can ostensibly debate and influence policy on all of them, because the policy itself is not secret.
But when even the policy is secret, it is totally beyond even the possibility of democratic control.
My favorite example of why the government shouldn't be allowed to keep that many secrets is the Supreme Court case which established the concept of "National Security" (secrets). It was a wrongful death lawsuit regarding a plane crash just after WWII (if I recall correctly). The government said, "we can't release any details of the crash because it's a matter of National Security". SCOTUS said fine and the families of the men killed in the crash were told to pound sand.
Jump forward to the modern day, when the details of the crash are accidentally released with a bunch of other declassified information. It turns out that the flight had absolutely nothing to do with "National Security", it was just a routine flight in a plane with a poor maintenance history. The government was just covering their asses to avoid paying out for wrongful deaths, and bullshit so hard they created a legal precedent.
Fun fact: the PAL codes on Strategic Air Command's Minuteman nuclear missiles were set to 00000000. To quote President Skroob, "That's amazing - I've got the same combination on my luggage!".
To some extent I'm not sure the "launch codes" themselves -- if there is something you need to type into a console at the missile base itself to actually launch the missile and there isn't just a shiny red button -- need to be that secret. There should be a boatload of physical security protecting nuclear missiles. If there isn't, we're doing it wrong. It's not like we should be letting random people reach the point they could fire the missile.
But launching nuclear missiles is a command decision so the weak point in the system is the means by which an order to launch is authenticated. And whatever secret tokens are used in that process, those I want as few people as possible to know (ideally zero).
Also, evidence obtained in a way that violates Constitutional amendments and that is therefore barred from admission in a case. Medical records (incl. genetic testing results). Tax returns. Information about a person inadvertently collected pursuant to a valid investigation, or otherwise not relevant to the investigation.
