Satyricon, written by Gaius Petronius of Nero's court and narrated by Encolpius, is a fragment. The section featuring Encolpius' adventures is fragmented, with no ending: "the exit of the two principal characters is not fixed at the time our fragments come to an end" (W. C. Firebaugh, Translator). Consequently, there is some difficulty in attributing purpose to Encolpius' narrative and actions.
If it is true, however, that Petronius' story had a deep and lasting influence on literature, as Firebaugh posits ("[Satyricon's] powerful influence upon the literature of the world"), then we ought to be able to borrow a literary model and a Shakespearean technique to overlay what we do have in order to discover a purpose attributable to Encolpius.
The literary model is that of laying out in the early pages of a work the key elements that will take shape through the course of the narrative. The Shakespearean technique is that of putting wisdom into the lines of seemingly insignificant characters and speeches: Shakespearean Clowns and Fools often carry the lines, in seemingly insignificant speeches, that establish key elements of plot, characterization and theme; King Lear's Fool comes readily to mind.
By using this approach—by borrowing the model and technique as an overlay—we can say that Agamemnon's recitation of verse in the fifth paragraph (called Chapter the Fifth)—being a foundational speech early in the fragment and being seemingly insignificant verse—holds the clue to both the direction of the plot development and to Encolpius' purpose.
We find in Agamemnon's recitation that he warns young men against "riff-raff," "evil companions" in dining and drinking, "poetry" and "sirens." We go on to find through Encolpius' narration that these are precisely the things that Encolpius, Ascyltos and Giton pursue, engage in and run from in their adventures. We also find in Encolpius' opening speech that he decries false rhetoric of "empty discord" and admonishes a "dignified ... a chaste, style" that "rises supreme by its own natural purity" (Chapter the Second), which he associates with the learning and wisdom of Sophocles, Euripides, Plato and Demosthenes.
When we put these two together—Agamemnon warning against the things that deter a young man from acquiring clear thinking and Encolpius admonishing "supreme" rhetoric—we can deduce that a reasonable purpose attributable to Encolpius would be that of illustrating the need for pursuit of wisdom, such as Agamemnon admonishes, through the pursuit of clear thinking expressed in "supreme" rhetoric, such as Encolpius admonishes.
If this suggested purpose proves plausible, then we might expect the story would have ended, being a satirical comedy, with Encolpius, Ascyltos and Giton returning to where they began, sadder but wiser young men, having indulged in all Agamemnon warned against and now newly devoted to the right course, that which was followed by Socrates, Demosthenes and Cicero.
Agamemnon: ...later, [after] the lore
Of Socrates' school he has mastered, the reins let him fling,
And brandish the weapons that mighty Demosthenes bore.
Then, steeped in the culture and music of Greece, let his taste
Be ripened and mellowed by all the great writers of Rome.
At first, let him haunt not the courts; let his pages be graced
By ringing and rhythmic effusions composed in his home [not to the Court]
In eloquent words such as undaunted Cicero chose.
Come! Gird up thy soul! Inspiration will then force a vent
And rush in a flood from [the] heart... (Chapter the Fifth)