Are there any state secrets the president would not be allowed to know if he asked?

This is a perennial question, actually. See:

Is there anything the President can’t know? (from 2002)
What is the POTUS’ level of clearance in top secret matters? (from 2008)
Are there any government activities so secret that even the president doesn’t know about them? (May 2011)
Does the President still operate on Need To Know? (August 2011)
What if the president elect were ineligable for a security clearance? (2013)
Trump and a POTUS security clearance. (March 2016)
Could a US President be denied security clearance? (May 2016)

I see no reason to think the consensus of those threads has changed, regardless of who is in the White House: No member of the executive branch can, Constitutionally speaking, deny to the President of the United States any particular “state secret” (not even “a list of all US spies”) on grounds of “need to know”. Congress, by statute, has put some additional restrictions on things like the information in individual tax returns. However, under the Constitution the President of the United States is head of the executive branch of the federal government, and commander-in-chief of the armed forces; no one within the executive branch (including the armed forces) can have constitutional authority to tell the President of the United States “I’m sorry, sir, you’re not cleared for that information”.

To quote myself from the dim, misty days of 2011:

But in terms of the law (including the supreme law of the land, the Constitution of the United States), the President of the United States has no “need to know” restrictions regarding “state secrets”.