The portion of this excerpt that includes a hyperbole is: "the Government of the greatest Nation on earth."
In oratory, a hyperbole is the use of exaggeration in order to inspire strong impressions or feelings in the audience. The sentence itself might not be or sound absolutely true, but it does evoke emotion and touches the listeners.
Here, we see an affirmation made by President Lyndon Johnson that the United States are the greatest nation in the world. This was said in his speech defending civil rights, in which he claims that there is no use in conquering the stars if Americans cannot respect each other and accept equality. By stating that they are the greatest nation, he is evoking feelings of patriotism, surely in hopes of having the audience connect such feeling to the topic discussed.
