That’s for the American people - through their elected President - to decide. Nothing is immune to oversight.
More likely, it’d be something like "Sir, are you absolutely certain you want to know the name of the asset we have in the Kremlin? As things stand now, there’s no way you can inadvertently reveal his name. "
The POTUS is certainly cleared for all classified information, regardless of its level of classification. He has the authority to decide if he has a “Need To Know”. Why a POTUS would need to know the value of resistor #12 of the guidance package for a Minuteman missile is beyond me, though.
It’s a 5kΩ pulldown resistor. I’d tell you more about it but there are men in black suits knocking on my door.
The President can’t know everything, for the simple reason that it would take many lifetimes to give him or her all that information and longer for him or her to absorb it.
But there cannot be a piece of information which a low-level employee could keep secret from the President, because no low-level employee could possibly understand all the reasons that the President had for asking about it. That kind of information flows upward, not downward.
Perhaps you may be thinking of the administratively named Yankee White (Wikipedia), and documented in DOD 5210.87 from 1998 (PDF Warning).
Perhaps. smithsb actually gives the proper context to my fuzzy post. At the time I was only familiar with Confidential/Secret/Top Secret rankings and this was 15+ years ago. With respect to “SCI” (plus “Poly”), I was unintentionally aware of these acronyms as a result of parsing through the Washington Post want ads. These jobs never held any appeal because they mostly involved managerial positions, so I never bothered to figure out what the stood for.
By the way, having those clearances doesn’t necessarily mean that your smarter for it; I found this link to be interesting.