Another reply on behave of a Fool
Before we can give a detail criticism to Anselm's argument, we want to define two terms: description and concept. By a description, we mean any expression that use to describe "something", e.g. a round table. By a concept, we mean a (logically) valid description, e.g. a circular square is not a concept but is a description. A description can be thought of (or conceived) only if it is valid (or is a concept), e.g. we cannot think of a color that is both red and blue. This distinction between description and concept can also help to clarify the answer to the famous question: "can God create a stone that is heavier than He can carry?" (see the endnote).
Anselm's argument goes as follows:
As other people, our major criticisms are on points (1) and (2). Let us use T to denote the statement: "the thing that than which nothing greater can be conceived". Some people doubted the validity of T as a definition for God. In fact, it is even doubtful whether T can be a definition for anything at all. T is certainly a description but it is not clear whether it is a concept. Russell pointed out in his famous paradox that the description: "the set of all sets" is not a valid one (this leads to a paradox). Therefore, we need to pay attention whenever we talk about statements that refer back to themselves or claim some sort of universality.
Secondly, some people disagree that "something that cannot be thought of as non-exist" is greater than "something that can be thought of as non-exist". In fact, there cannot be any order between these two statements at all. In the case of T, even though if we assume that T is a concept, the statement "T can be thought of as non-exist" and the statement "T cannot be thought of as non-exist" is mutually exclusive. Hence if one of these statements can be thought of, than the other one cannot be thought of. Therefore, there cannot exist an order between them to tell which one is greater than the other.
Another not so important challenge to Anselm's argument is that even if the description T exists in reality, there is no indication that such thing should be unique. As in Mathematics, there can be more than one maximal elements for a system.
Endnote
The question: "can God create a stone that is heavier than what he can carry" was used to challenge the concept of the Omnipotent of God. If we can show that under the assumption of God's Omnipotent, we can still give an answer to this question (in fact, a negative one) such that no contradiction arises, then we are done. Now if we assume that God is Omnipotent, then the description: "a stone that is heavier than what God can carry" is logically invalid. If God cannot do something that is logically invalid (e.g. to draw a perfect circle on a sheet of paper which is at the same time a square), then we cannot say that this negates the Omnipotent of God. Therefore, answering the original question negatively will do.