According to popular notion of Deep State in the United States, it is the civil servants, particularly the intelligence agencies' influence over elected officials. https://en.wikipedia.org/wiki/Deep_state_in_the_United_States
But is it really possible that elected officials who have the power to fire heads of these civil servants and intelligence agencies become subservient or helpless against such Deep State? Through chain of command the President or his appointed officials can get any employee lower down the chain fired. If an agency was created by the President, the President can dissolve that agency. If the agency was created by Congress it can be dissolved by Congress.
So it is extremely fishy that a powerful Deep State exists which has power over the President and Congress. Perhaps there is a power above the President and the Congress which protects and leads the Deep State? Something that stays constant across different Presidents, just like the civil services and intel agencies stay constant across different Presidents of the United States, but which is above the president rather than below.
For the answer let's look at the secret agency CIA (Central Intelligence Agency) which is considered a major stronghold of Deep State. Let us look at CIA's history. CIA was the peace-time successor to war-time OSS (Office of Strategic Services). How did OSS originate? Let us look at Origin section for OSS's Wikipedia entry https://en.wikipedia.org/wiki/Office_of_Strategic_Services#Origin
President Franklin D Roosevelt was concerned about American intelligence deficiencies during WW2. On suggestion of William Stephenson, the senior British intelligence officer in western hemisphere, plan was drafted for an intelligence service based on model of British Secret Intelligence Service (MI6) and Special Operations Executive (SOE). The plan was drafted by William J. Donovan. He was appointed coordinator of information heading the new organization known as office of coordinator of information.
The organization was developed with British assistance. Donovan had responsibilities but no actual powers. Until some months after Pearl Harbour, the bulk of OSS intelligence came from UK. British Security Co-ordination (BSC) trained first OSS agents in Canada, until training stations were set up in US with guidance from BSC instructors who also provided information how SOE was arranged and managed. The British immediately made available their short-wave broadcasting capabilities to Europe, Africa and Far East.They provided equipment for agents and American production was established. The entry for MI6 in Encyclopedia Britannica also boasts that MI6 helped to train the personnel of US Office of Strategic Services and has since co-operated with its successor CIA. https://www.britannica.com/topic/MI6 Thus the creation of OSS was suggested as well as executed by the British. Britain is of course monarchy so this was all with the blessings of British Monarch.
So the real question is, why would British Monarchy approve of creating a powerful agency in another country, just because they are war allies? US hadn't even joined WW2 yet when OSS was created. The obvious answer is that British Monarchy felt assured that the OSS or the CIA would never be used against itself. The reason for such assurance would of course be that they controlled USA. And they still do, through Deep State.
US Presidents come and go but British Monarchy is constant. Thus our requirement for deep state that it should be above the President yet constant is clearly met too!