Distinguishing an object and a separate noun

I'm having a hard time seeing how nouns become the object for a verb and sometimes how they aren't. 

For example, some objectless clauses (From http://dictionary.reference.com/search?r=8&q=verged):

a statesman who verged on greatness - Here, greatness is a noun and the verb 'verged' is acting on it.

a situation that verged on disaster. - Here, disaster is a noun and the verb is doing the same thing.

However, according to http://en.wikipedia.org/wiki/Object_%28grammar%29

Now, in the clauses from the dictionary, there is the preposition 'on' and the nouns 'greatness' and 'disaster'.  What I don't get is how 'on greatness' and 'on disaster' aren't objects.

The mistake you're making here is that you are assuming "greatness" and "disaster" are both functioning as nouns in these sentences.  They are not.  To understand a word's 'part of speech,' you must see how it functions in the sentence.  The same word can have a different part of speech in different sentences, depending on how it interacts with the other words of the sentence in question. 

The two clauses in your first two sentences contain 'adjective clauses.' These are clauses that modify nouns and pronouns.  Most adjective clauses start with 'relative pronouns' or when, where or why.  Relative pronouns include who, that, which, whom, whose, whoever, whomever, whichever, what and whatever.  Adjective clauses beginning with a relative pronoun may also be called relative clauses.

Your two adjective clauses are:

"who verged on greatness."  (describing the antecedent "statesman")

and

"that verged on disaster."  (describing the antecedent "situation")

Those clauses each contain the preposition 'on,' which combines with the following words to form a 'prepositional phrase.'  The prepositional phrases of each clause are:

"on greatness"

and

"on disaster"

Prepositional phrases are units that function as an adjective or an adverb.

Here, "greatness" and "disaster" are functioning as adverbs as part of their respective prepositional phrases "on greatness" and "on disaster."  Once again, just because a word is one part of speech in one sentence, it doesn't mean it has the same part of speech in every sentence.  Ain't grammar fun?