I contend that the concept of an objectively Greatest Possible Being ("GPB" for short) isn't a coherent concept. As the concept is about being greater than other things, we're talking about Partially Ordered Sets ("posets" for short).
To argue a fortiori, I will be making the following GPB friendly assumptions:
0) The greatness interval is bound for all Great Making Property ("GMP" for short) orderings.
1) The orderings for all GMP are chains (totally ordered).
2) The greatness orderings for each GMP are objective. There is an objective fact of the matter that more moral is greater than less moral.
3) There are objective GMP. There is an objective fact of the matter as to whether a given property is a GMP.
For the beginning, we'll stick with independent GMP to make things easy and to clearly illustrate the problem.
Let's look at two great-making properties, P and Q.
The value ordering of P is <P, <> = P1 < P2 < ... < Pn.
The greatness ordering of P is <P, ≺> = P1 ≺ P2 ≺ ... ≺ Pn.
Similarly, for Q, we have both value ordering and greatness ordering.
The value ordering for Q is <Q, <> = Q1 < Q2 < ... < Qm.
The greatness ordering for Q is <Q, ≺> = Q1 ≺ Q2 ≺ ... ≺ Qm.
Where "X < Y" is "the value of X is less than the value of Y" and "X ≺ Y" is "X is less great than Y".
Now, consider two beings, A and B, who exemplify both P and Q to varying degrees. Being A exemplifies P7 and Q12, while entity B exemplifies P12 and Q7.
Of the entities A and B, which is greater?
To answer that question, we need to look at the product space: PxQ. That's the set of all possible combinations of values of P and Q.
So, the entity A, on PxQ, corresponds with point (P7, Q12), and, likewise, B corresponds with point (P12, Q7).
Even with the objective ranking, it's not possible to give an objective answer. The product ordering on PxQ only gives a partial ordering, and it's one such that there is no answer as to which of A and B is the greater being [(A ≺ B ) iff ((P(A) ≺ P(B)) and (Q(A) ≺ Q(B)))].
So, the only way to compare them is if one entity is greater in both properties than the other entity. That makes tons of entities not directly comparable.
Each added GMP makes more entities incomparable. If we have three GMP, than one entity is greater than another only if it is greater in terms of all three GMP.
At this point, you might be wondering, "So? GMP has the property value corresponding to the greatest greatness for all properties. It's (Pn, Qm).". And, if we only had independent properties to deal with, you'd have a point. I introduced independent properties first, so you could see that this is a problem with *ALL* GMP. Not all, GMP, however, are independent. The values of some GMP are linked to the value of other GMP. Sometimes, the more one GMP is exemplified, the less another is.
If we then move on to great-making properties such that they aren't independent, but are rather somewhat inversely related (such as moral goodness and potence), then you can't max out the product ordering, since raising one property lowers the other.
They come in pairs:
(P1, Qm), (P2, Qm-1), ... , (Pn-1, Q2), (Pn, Q1)
There is no place in this space of property pairs where one entity is greater than another in all properties. Thus, when we introduce inversely related GMP, we go from losing some ordering to losing all ordering.
There is no objective ordering such that a GPB exists.