Are there any government activities so secret that even the president doesn't know about them?

It is perfectly clear and incontrovertible that the President is in charge of what goes on in the Executive Branch. The Constitution says so.

How the government keeps secrets is a matter that is primarily regulated by a series of Executive Orders. In short, the “need to know” is based on a Presidential order, and as the chief executive, there is no higher authority in government on who can determine need to know. All classification authority derives from the President, so barring a “can God make a burrito so big that He cannot eat it” type of metaphysical question, of course the President is entitled to knowledge of all classified information.