Since some here seem to have more opinion than research, I'll clarify a couple things.
This is not based on the security system directly, but things ranging from nice books like one about declassified 'Presidents' eyes only' documents called "For the President's Eyes Only", James Bamford's books, histories by the people who have worked with Presidents on confidential information and other things.
The President has no formal limits I'm aware of, he has, as he should, 'full clearance'.
Now, there are informal limits to that. The President is normally never given things like sources' names, that are scrubbed out of reports for him, as he has no 'need to know' them. The President cooperates with this. In theory, if he wanted a name, he could demand it, and the staff could either supply it or resign/be fired, but it doesn't happen.
With an exception like Valerie Plame, who got some attention from Cheney's office.
Another informal limit is simply people who hide things. They're not supposed to, they do.
No President was able to tell J. Edgar Hoover 'give me all the dirt you have'. Intelligence contacts with Lee Harvey Oswald were kept quiet for some time, for example.
To this day, there is a lot of evidence suggesting more of a relationship between the government and Oswald that has never been made public, to Presidents either.
Presidents do leak to the press for their own reasons. It not only ingratiates them with the reporters and gets them owed favors, it helps get their message across.
Watch for "A senior administration official" or "A top White House official" type descriptions of a source, they've repeatedly meant the President himself.
Indeed, the President is more able to use leaks for his own reasons while people below him may do so but usually risk the anger of the President or more.
The Daily Briefing released for the day Bush was told 'Al Queda wants to attack' was an example of the level of information in that briefing.
JFK in particular used to regularly call low level people in the government for information because he felt he wasn't getting what he wanted from higher officials.
A Sergeant's phone would ring and he'd be told the President wanted to talk with him.
Finally, Nixon did something bad on this, tried to lie about JFK to the media, saying he'd found documents in a safe implicating JFK in the Bay of Pigs on issues.
Remarkably, the press determined the fraud and it didn't work.