US Presidents receive their clearances as a result of their position. At least a couple of Presidents might not have qualified for (or might have lost) some of their access if they had to qualify via the traditional investigation, due to associations with known or suspected foreign agents (Kennedy), drug use, bankruptcy (which Truman barely avoided) or many other causes. Of course, there is some flexibility (and even more inconsistency) in how the standards are applied, even in ordinary clearance investigations.
Presidents aren’t automatically cleared for anything they feel like asking. As noted by others, above a certain level, information is on a “need to know” basis, and there are many things a President doesn’t need to know. The classic example is “source identity” info. S/He may need to know that we have a proven reliable source inside <organization of your choice> who has told us X, Y and Z, but s/he almost certainly does not need to know the name of that source.
“Means and methods” is traditionally among the most closely guarded classes, with source identity being a particularly sensitive subset - even accidental hints about means, methods and sources can help the target close them down or at least impede their use. [e.g. many of our classic adversaries might execute a ‘suspected traitor’ on little more than a suspicion, and even when they get the wrong guy, it’s likely to make the real source wary. Similarly, some access to electronic comms could be thwarted or impeded if their nature or even existence was suspected.]
The question has been asked (in another thread): “Who tells the President he can’t know?” Well, as a practical matter, Presidents don’t just walk into the agency of their choice and grab files. They are briefed by intelligence experts in each given field. There is some filtering at that level, which can be colored by instructions by the expert’s superiors (and, as noted scandals over the past 40 years, sometimes additional undesirable filtering due to organizational agendas)
As a practical matter, Presidents understand this (it’s part of their briefings from the time they become serious candadates) and, offhand, I can only think of one case when a President really fought it. I’ve heard thirdhand accounts of Presidents being less-than-happy, or even suspicious, about the info in their briefings, but I don’t know how accurate those reports were, and such incidents are probably inevitable anyway, even withthe most well-meaning of briefing experts.
When if push ever came to shove on such a conflict, it would probably be determined by backstage political infighting, rather than a Supreme Court ruling, explicit Executive Branch firings or any other public event. (Hey, it’s Washington. What’d you expect?) I would guess the infighting starts well before the crisis level, and probably at much lower levels than the President himself.