Possible world: A state of affairs which is different from the existing world. (The only prerequisite is that it cannot contain a logical contradiction, like a “married bachelor”.)
Possible existence: A being that exists in some, but not all possible worlds.
Necessary existence: The being exists in all the possible worlds.
In no world can a logically contradictory state of affairs exist.
To show that there is a “necessarily existing being” one must examine all the (infinitely many) possible worlds, and find out that this being is present in all of them. That is impossible. On the other hand it is very simple to prove that there is no “necessarily existing” being. All we need is to find is two possible worlds without intersection, and this proves that there can be NO necessarily existing being. Go, and find two such worlds. It is easy!
The concept of “maximal greatness” is incoherent. The author picked a few “good sounding” (but pretty much nonsensical) attributes, and declared that these attributes are necessary and sufficient for a being to become “maximally great”. It is impossible to find a maximally great “dinner”, or a maximally great “vacation”.
The concept of “possibly necessarily” is even worse. If something is “necessary”, then it is also “possible”. But from the “possible” there is NO way to “necessary”, S5 modal logic or otherwise.