The crossword solution for the term BE EXTREMELY EAGER is ACHE, which is a four-letter word. ACHE is a verb that means to feel a dull, continuous pain or to feel a strong desire or longing for something. It can also be used to describe a feeling of intense eagerness or enthusiasm. Therefore, the phrase BE EXTREMELY EAGER can be summed up as 'to feel a strong desire or longing for something with great enthusiasm.'