The following is a version of the Anderson-Goedel ontological argument with somewhat weaker premises, and which is apparently not subject to Oppy's parody (Analysis, 2000):
Axiom 1. If a property is positive, its negation is not positive.
Axiom 2. If P is a positive property, and P entails Q, then Q is a positive property.
Axiom 3. Necessary existence is a positive property.
Definition 1. If P is a property, then EP is the property of having P essentially.
Definition 2. P is strongly positive iff it is necessary that EP is positive.
Definition 3. God-likeness is the property of having all strongly positive properties.
Axiom 4. God-likeness is a positive property.
Given an appropriate modal logic that includes S5 and assumes that properties are necessary beings, we get:
Theorem 1. If Axioms 1-4 hold, then there necessarily exists an essentially God-like being.
Note: Since EP entails P, by Axiom 2, any strongly positive property is positive.
The details of proof don't matter much but are below the fold for completeness.
The hard question is whether we can give an interpretation to the notion of a "positive property" that makes Axioms 1-4 plausible and the conclusion of Theorem 1 interesting.
The proof depends on several subsidiary results. (I start with Lemma 2, because Lemma 1 turned out to be unneeded and I don't want to renumber.)
Lemma 2. Given Axioms 2 and 3, necessary existence is strongly positive.
Proof of Lemma 2: Let N be necessary existence. If x exists necessarily, then it is necessarily true that x exists necessarily, by S4. Hence, N entails EN. By Axioms 2 and 3, EN is positive. Moreover, this proof works necessarily (we only made use of the axioms, and axioms are assumed to hold necessarily), so EN is necessarily positive.
Lemma 3. Given Axioms 1 and 2, every positive property is possibly exemplified.
Proof of Lemma 3: Suppose for a reductio that P is positive but not possibly exemplified. Hence, necessarily, nothing has P. Hence, necessarily and trivially, everything that has P has ~P. Hence, P entails ~P. Hence, ~P is positive as P is, by Axiom 2. But ~P is not positive, by Axiom 1. Hence absurdity ensues.
Lemma 4. If Axiom 2 holds, and P is strongly positive, then EP is strongly positive.
Proof of Lemma 4: Necessarily, EP is positive. EP entails EEP by S4. Hence, necessarily, EEP is positive, by Axiom 2. Hence EP is strongly positive.
Lemma 5. If P possibly is strongly positive, then P is strongly positive.
Proof of Lemma 5: Suppose possibly P is strongly positive. Then, possibly, necessarily EP is positive. By S5, necessarily, EP is positive. Hence, P is strongly positive.
Lemma 6. Given Axioms 1 and 2, if x has God-likeness, then x essentially has God-likeness.
Proof of Lemma 6: Suppose for a reductio that x possibly lacks God-likeness. Then, possibly, there is some strongly positive property that x lacks. Since properties are necessary beings, and given S5, existential quantification over properties and possibility can be interchanged. Hence, there is a property P such that possibly P is a strongly positive property that x lacks. Let P be such. Then, possibly P is strongly positive. Hence, P is strongly positive by Lemma 5. Hence, x essentially has P. But we said that possibly x lacks P. Hence a contradiction ensues.
Proof of Theorem 1: God-likeness is possibly exemplified, by Axiom 4 and Lemma 3. Thus, possibly, there is an x that has God-likeness. God-likeness entails necessary existence, by Lemma 2. Hence, possibly, there is a necessary being x that has God-likeness. But, by Lemma 6, anything that has God-likeness has God-likeness essentially, and this holds necessarily since it is a logical consequence of the axioms. Thus, possibly, there is a necessary being that essentially has God-likeness, since if a necessary being has an essential property, it is necessary that it have this essential property. Hence, possibly, necessarily there is an x that is essentially God-like. Hence, necessarily, there is an x that is essentially God-like, by S5.