The sentence is: No one grew tired of eating it. Is there any prepositional phrases in there?

The other sentence says: Linking verbs link the subject with a noun, a pronoun, or an adjective. Is everything after with(also including with)a prepositional phrase?

Although your first sentence looks like it has a prepositional phrase, tired of is a verb phrase.
http://dictionary.reference.com/browse/tired%20of

Yes. The prepositional phrase in the second sentence is with a noun, a pronoun, or an adjective.

In the first sentence, "Of eating it "is a prepositional phrase. "of" is the preposition and "eating it" is the gerund object of the prep.

To determine if a sentence contains a prepositional phrase, you need to identify the preposition and the object of the preposition. In the sentence "No one grew tired of eating it," the prepositional phrase is "of eating it."

Here's how you can identify it:
1. Look for a word that shows a relationship between two other words or phrases. In this case, the word "of" is the preposition.
2. Identify the noun or pronoun that follows the preposition. In this case, "eating it" is the object of the preposition.
3. Determine if the object of the preposition can be modified by words like adjectives or adverbs. In this case, "eating it" is a gerund phrase functioning as a noun and cannot be modified further.

Therefore, "No one grew tired of eating it" contains the prepositional phrase "of eating it."

Now, let's apply the same process to the second sentence. "Linking verbs link the subject with a noun, a pronoun, or an adjective."

The prepositional phrase in this sentence is "with a noun, a pronoun, or an adjective." Here's how you can identify it:
1. Look for a word that shows a relationship between two other words or phrases. In this case, the word "with" is the preposition.
2. Identify the noun or pronoun that follows the preposition. In this case, "a noun, a pronoun, or an adjective" is the object of the preposition.

Therefore, "Linking verbs link the subject with a noun, a pronoun, or an adjective" contains the prepositional phrase "with a noun, a pronoun, or an adjective."