Does one US military branch think it's superior?

My experience is no. I work in a joint command with a Marine Infantry Officer as my direct supervisor with a Navy Surface Warfare Officer (SWO) as our element's vice commander and an Army Ranger as our element lead. Our whole organization is commanded by a USAFR B-52 pilot.

In that experience I have never got the vibe so far that one is better than the other. Each service provides a specific function that is integral to a mission. The Marines can do a lot initially without the AF or the Army. But chances are that they get crucial intel from all of the above. The MEU's are poised for rapid response, but the Army is poised for long term engagement. The Air Force provides a good deal of long range persistent surveillance as well as the ability to conduct deep strike interdiction with our B'2s. In addition the AF provides crucial rapid global mobility mobility capabilities.

In the joint planning world I have the opportunity to see where all of the pieces fit together. It is never a one size fits all or else we wouldn't have a job. We find that in most cases to solve a problem it involves the engagement of several parts that include the whole spectrum of the DOD's capability.

Of course. More boats than the Navy, more aircraft than the Air Force, more Infantry than the Marine Corps...the Greatest Fighting Force the world has ever seen!

Every branch thinks they're superior in some way, shape, or form. And in some aspects they are. USAF in cyber and being stuck up, Army in fuck fuck games, Marine Corps in ground warfare and dicking around, and Navy in floating endless across the seas. (Sir).

Working in Veteran services at my college i would say yes there is. That being said we all respect the other services and know they have a different mission. But being in an office full of Army and Marines I do here them rag on my Air Force all the time. They do cut me some slack though since I was in AFSOC which meant my boss was still a 4 star Army general.

I agree.

It is a good fun rivalry between the services. When it comes to our beloved nation all services on the same page.

There is always going to be a rivalry among the branches, especially among lower enlisted. As I served with the Marines in Camp Leatherneck, Afghanistan, this became more apparent. I was able to get along with the officers within the Marines and they even let me play on their softball team during a stand down day. While, just about every time an Army Soldier had to interact with a Marine to provide logistical support there were words exchanged about the rival branch. One time it even came to blows.
I started out as a private in the Infantry and I can see why there is such a passionate and uneducated view of the other branches. We were indoctrinated in basic training to believe we were the baddest MoFos on the planet, and so are Marines, Navy, and even the Air Force (ok, maybe not the Air Force...just kidding). That 9-13 weeks of complete submersion in the propaganda is tough to get out of your head, but as we are exposed to more Joint Operations, it is easy to see (if we let our guard down) that all branches have some pretty amazing service members doing some pretty amazing things to protect this country.
Perspective is hard to change if you don't allow yourself to move to a different viewpoint.

While each service likes to boast they're better than the others, the Marines are the only ones who actually believe it.

If you have pride you are always going to have a sense of inner service rivalry as much as there is one between Cavalry and Infantry units within the US Army. On the other hand, I think for those who have deployed to a war-zone and been in combat-it is less of a thing that matters to people. When it comes between the Army and Marines-the most gung-ho people tend to be the ones who never saw a combat deployment. At least this has been my experience and my perception.

There's definitely a rivalry but it's all in good fun.