There's actually one more branch, the Air Defense (EAD) which is huge, and some might even consider it to be the Egyptian military's strongest asset since its performance has been exemplary since the military instilled it as a major and also independent branch of the Egyptian military after the disaster of 1967 and knowing they were going to be going to war for a long time, starting with the ensuing war of attrition.
Following the 1967 war, Egypt took the decision to reorganize and develop its armed forces and included those decisions on Presidential Decree No. 199 issued on 1 February 1968, establishment of the Egyptian Air Defense Forces as a separate branch, standing alone, avoiding the previous weapons and units scattered among departments. With artillery and rocket troops were tracing artillery management units and radar warning and operations centers used to belong to the air force and the control points were given follow the Border guard.
Great thread on how extensive it is and how it's tied in with the other branches as it operates as part of the entire defensive network
HERE.
But to answer your question, I don't think there really is one that's stronger than the other like in other countries. I think with Egypt, the weight & responsibility of each branch is very heavy to perform at the highest level because the others depend on it, with maybe the exception of the Navy to a certain degree. For example, the Air Defense has to protect all the others, from the air force's airports to the armored columns to radar installations to naval & submarine ports even certain critical civilian infrastructures.
At the same time, it relies on the air force to intercept unwanted violating intrusions and to conduct unknown presence of ground, ocean or aerial targets that haven't violated air or waters or land but are approaching them.
The Air Force has a lot of duties, mainly patrolling & ready at all times for interceptions and war. It's always conduction patrols & recon missions all over the EGY territory from the northern Mediterranean Sea coast to the western Libyan border all the way south to the Sudanese border and up the entire easter Red Sea coast to Sinai and especially the Suez Canal. If you're even at any spot along the canal your chances of seeing jets fly by are pretty high.
The EAF also works hand in hand with the EADS and quite extensively as a matter of fact since the former also relies heavily on the latter with the RICS-4/5 network that they operate under. This is the Egyptian Military's domestic version of network-centric environment which they've taken very seriously since the mid-70s but have really perfected it in the last 15 years or so, especially once the Rafale was inducted into the EAF.
The navy obviously is tasked with huge responsibilities being it has the entire Mediterranean Sea coast and all its offshore gas assets and major ports such as Port Said, Alexandria, Marsa Matruh, Port Tawfik, Bernice, Hurghada, Taba, Sharm El Sheikh, El Arish, and of course, the Suez Canal itself and all offshore oil rigs in the Red Sea as well as transport shipping in and out of the Gulf of Suez. So it has its hands full for sure and is composed of a huge fleet of ships including the two Mistral LHDs & 8 submarines 4 of which are older but still work well and more to come soon. The navy also has a lot of its own defensive capabilities, but its ports rely on both, the EADS' protections as well as the EAF's. There's also the Naval Paratroopers division and the Egyptian version of SEALs (frogmen division) and other naval special forces.
The army with all the armor from the huge inventory of tanks, APCs, IFVs, artillery of all sorts also relies on mobile SAM units from the EADS to be attached to its divisions. That adds to the weight & responsibility of the EADS additionally. The army's armored units bear the responsibility of being ready at a moment's notice for war. Other than that, there really isn't much else required of them, as in constant duties during peace time such as those of the EADS (explained), the Air Force or the Navy but then there is the huge army which comprises of many large outfits such as the Sa'aka (Thunderbolts) the Rapid Deployment Forces, the huge membership of regular infantry and of course, a large makeup of special forces units that are actually interchangeable with some of the other branches such as the armor & navy. All of these have also been engaged in many of the counter-terrorism missions in the past 10 years.
So, as you can see, they all have their own very extensive responsibilities and are quite interconnected in many ways so they all essentially depend on one another to some degree which forces each's strength to be as strong as it possibly can to make that ever so important network-centric environment work well. Hard to label one as the strongest than the others but based on all the above info and if I was forced to pick one, I would say it's the Air Defense branch. Its responsibility for most of the others as well as civilian infrastructure is ginormous along with its historical experience, I would have to say makes it the strongest of the lot.
BTW, there's also Egypt's underground ballistic missile program that not many people know about including the CIA which has tried time & time again to spy & infiltrate but to no avail. I'm not sure where they would categorize that force, with the Air Defense? Or with the Army? Something interesting to think about.